Metaphor and Mathematics
A Thesis Submitted to the
College of Graduate Studies and Research
in Partial Fulfilment of the Requirements
for the degree of Doctor of Philosophy
in the Unit of Interdisciplinary Studies
University of Saskatchewan
Saskatoon
By
Derek Lawrence Lindsay Postnikoff
c
Derek
Lawrence Lindsay Postnikoff, April 2014. All rights
reserved.
Permission to Use
In presenting this thesis in partial fulfilment of the requirements for a Postgraduate degree
from the University of Saskatchewan, I agree that the Libraries of this University may make
it freely available for inspection. I further agree that permission for copying of this thesis in
any manner, in whole or in part, for scholarly purposes may be granted by the professor or
professors who supervised my thesis work or, in their absence, by the Head of the Department
or the Dean of the College in which my thesis work was done. It is understood that any
copying or publication or use of this thesis or parts thereof for financial gain shall not be
allowed without my written permission. It is also understood that due recognition shall be
given to me and to the University of Saskatchewan in any scholarly use which may be made
of any material in my thesis.
Requests for permission to copy or to make other use of material in this thesis in whole
or part should be addressed to:
Dean
College of Graduate Studies and Research
University of Saskatchewan
107 Administration Place
Saskatoon, Saskatchewan S7N 5A2
Canada
i
Abstract
Traditionally, mathematics and metaphor have been thought of as disparate: the former rigorous, objective, universal, eternal, and fundamental; the latter imprecise, derivative, nearly — if not patently — false, and therefore of merely aesthetic value, at best. A
growing amount of contemporary scholarship argues that both of these characterizations are
flawed. This dissertation shows that there are important connexions between mathematics
and metaphor that benefit our understanding of both. A historically structured overview of
traditional theories of metaphor reveals it to be a notion that is complicated, controversial,
and inadequately understood; this motivates a non-traditional approach. Paradigmatically
shifting the locus of metaphor from the linguistic to the conceptual — as George Lakoff, Mark
Johnson, and many other contemporary metaphor scholars do — overcomes problems plaguing traditional theories and promisingly advances our understanding of both metaphor and
of concepts. It is argued that conceptual metaphor plays a key role in explaining how mathematics is grounded, and simultaneously provides a mechanism for reconciling and integrating
the strengths of traditional theories of mathematics usually understood as mutually incompatible. Conversely, it is shown that metaphor can be usefully and consistently understood
in terms of mathematics. However, instead of developing a rigorous mathematical model
of metaphor, the unorthodox approach of applying mathematical concepts metaphorically is
defended.
ii
Acknowledgements
The support I have received from my community of colleagues, family, and friends over
the course of this decade-long project has seemed non-compact — that is, without limit.
Many individuals deserve recognition for a variety of important contributions. My supervisor, Dr. Sarah Hoffman, has been unwaveringly encouraging; this document owes much to
her scholarly guidance and her generosity. I also owe a debt of gratitude to my advisory committee: Dr. Eric Dayton, Dr. Karl Pfeifer, Dr. Murray Bremner, Dr. Florence Glanfield, and
Dr. Emer O’Hagan. They provided many insightful suggestions, willingly took on the many
responsibilities associated with advising an interdisciplinary student, and were patiently supportive when personal crises slowed my progress. Thanks also to my external examiner, Dr.
Brent Davis, for his useful and flattering comments.
In addition to my committee, I wish to thank three individuals for contributing to the
content of my dissertation: Dr. John Porter (who provided invaluable assistance with Ancient
Greek); Dr. Ulrich Teucher (who introduced me to the work of Cornelia Müller); and Ian
MacDonald (who suggested I read Jesse Prinz). I would also like to thank my peers and colleagues in the Philosophy Department for valuable discussions and camaraderie, particularly
Aaron, both Wills, Leslie, Sean, Scotia, Kevin, Mark, Diana, Jeff, Ian, and Dexter.
A very sincere thank you to the support staff of the university — custodians, secretaries,
heating-plant employees, librarians, and so on — whose efforts to maintain our institution go
unnoticed far too often. Your work is appreciated! In particular, thanks to Della Nykyforak,
Debbie Parker, Susan Mason, and Alison Kraft for their immense help in navigating the
administrative aspects of my program.
This research was funded in part by various scholarships awarded by the University of
Saskatchewan; by employment opportunities given by the Philosophy Department, St. Peter’s
College, the Mathematics and Statistics Help Centre (Holly Fraser), The Ring Lord (Jon and
Bernice Daniels), and DAB Welding (Dave Bodnarchuk); and by subsidies from the taxpayers
of Canada. I am supremely grateful for these financial contributions.
Finally, this document would not exist without the emotional, financial, and intellectual
support of my family: thank you all!
iii
To Thora, point of my compass, patiently steadfast and
wholeheartedly supportive while I ran in academic circles.
iv
Contents
Permission to Use
i
Abstract
ii
Acknowledgements
iii
Contents
v
List of Abbreviations
vi
1 Introduction
1
2 A Philosophical History of Metaphor
2.1 Ancient Greece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Early Modern and Modern Philosophy . . . . . . . . . . . . . . . . . . . . .
2.3 Twentieth Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
9
23
32
3 Conceptual Metaphor
3.1 Theories of Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Conceptual Metaphor Theory . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Criticisms and Objections . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
59
69
96
4 Mathematics is Metaphorical
4.1 Traditional Theories of Mathematics . . . . . . . . . . . . . . . .
4.2 Conceptual Metaphor Theory and Embodied Mathematics . . . .
4.3 Yablo’s Mathematical Figuralism . . . . . . . . . . . . . . . . . .
4.4 Conceptual Metaphor Theory and the Philosophy of Mathematics
.
.
.
.
120
122
133
157
164
5 Metaphor is Mathematical
5.1 Computational Linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Mathematical Metametaphors . . . . . . . . . . . . . . . . . . . . . . . . . .
183
184
190
6 Conclusion
204
References
208
v
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
List of Abbreviations
BCE
CE
CMT
OED
UP
Before the Common Era
Common Era
Conceptual Metaphor Theory
Oxford English Dictionary
University Press
vi
Chapter 1
Introduction
Mathematics is the classification and study of all possible patterns.
Pattern is here used. . . in a very wide sense, to cover almost
any kind of regularity that can be recognized by the mind.1
— W.W. Sawyer
What cognitive capabilities underlie our fundamental human achievements?
Although a complete answer remains elusive, one basic component is a special
kind of symbolic activity — the ability to pick out patterns, to identify
recurrences of these patterns despite variation in the elements that compose
them, to form concepts that abstract and reify these patterns, and to express
these concepts in language. Analogy, in its most general sense, is this ability to
think about relational patterns. As Douglas Hofstadter argues. . .
analogy lies at the core of human cognition.2
— Holyoak, Gentner, and Kokinov
In the Western tradition, mathematics and metaphor have long been thought of as inhabiting opposite ends of the intellectual spectrum. Mathematics is generally considered the
paradigm case of rigor and objectivity. Mathematical theorems supported by valid proofs
seem to express truths that are exact, eternal, and evident. The precision and certainty
afforded by mathematical techniques underpin the extraordinary success of the quantitative
sciences. Even though many people find the practice of mathematics difficult and unenjoyable, most nonetheless acknowledge the contribution mathematics makes to the flourishing
of our species. Metaphor is generally not regarded so highly. At best, tradition regards it as
a convenient linguistic device for communicating subjective experiences (as in poetry) and as
a temporary measure when precise, literal language does not yet exist. At worst, metaphor
1
W.W. Sawyer, Prelude to Mathematics (Harmondsworth, Middlesex: Penguin, 1955), 12; emphasis his.
Keith Holyoak, Dedre Gentner, and Boicho Kokinov, “Introduction: The Place of Analogy in Cognition,”
The Analogical Mind: Perspectives from Cognitive Science, Ed. Gentner, Holyoak, and Kokinov (Cambridge:
MIT Press, 2001), 2.
2
1
is considered an unnecessary and avoidable figurative impediment to clear communication,
a mere step away from outright prevarication. Thus, metaphor has often been conceived as
antithetical to but also disparate from mathematics.
There are a variety of reasons to question this traditional view. For one, mathematics
and metaphor do not seem as disparate as suggested above. In my decades of experience
as a mathematics student, educator, and researcher, I have observed the use of metaphor
and analogy at every level of mathematical practice, from young novices learning to add to
conference presentations by Fields Medallists. Three specific examples will help substantiate this observation. First, when children are first learning addition their teachers draw a
comparison between the addition of numbers and the act of combining collections of physical
objects. Second, in order to help students understand the somewhat difficult notion of a
mathematical function, teachers and professors often describe functions metaphorically as
machines that take in numerical inputs and perform various manipulations and operations
upon them to yield numerical outputs. Third, some mathematicians (myself included) use
analogy in trying to understand multidimensional spaces by way of their experiences of the
three spatial dimensions they inhabit. For example, the vertices of the two-dimensional geometric object known as the Penrose tiling are sometimes understood as projections of a
five-dimensional cubic lattice in a similar way to how three-dimensional objects project twodimensional shadows.3 These examples are not isolated instances; many more mathematical
metaphors reveal themselves once one starts looking out for them.
Another questionable aspect of the traditional view is its suspicion of and hostility towards
metaphor. Historically, several authors have spoken out against this traditional dismissiveness, claiming that metaphor has been significantly undervalued and mischaracterized. These
scholars argue that metaphor should be embraced as a fundamental and pervasive part of human experience, not seen merely as an obfuscating derivative of literal language that should
be avoided whenever possible. The advances and increased interest in language scholarship
that occurred over the past century have correspondingly generated a substantial literature
on metaphor, much of which views metaphor in a more positive light than earlier works.
3
N.G. de Bruijn, “Algebraic theory of Penrose’s non-periodic tilings of the plane I,” Indagationes Mathematicae 84.1 (1981): 40.
2
Some contemporary authors even claim that metaphor is conceptual in nature and therefore
frequently precedes literal language rather than being derived from it. If metaphor is a basic
cognitive mechanism that helps structure our conceptual system then it is plausible that
metaphor could play a constitutive role in our understanding of mathematics. Even if one
rejects the idea that metaphor is conceptual, it seems that metaphor is a more complex,
legitimate, and widespread phenomenon than was previously suspected.
While scholarship of the last century generally improved metaphor’s reputation, it simultaneously brought mathematics down to earth a little. Mathematics was revered since the
time of the Ancient Greeks as the closest we flawed, mortal humans could come to knowing
objective Truth. The dramatic mathematical advances of the nineteenth century revealed
that mathematics is more varied, expansive, and complicated than was previously thought.
In particular, the discovery of non-Euclidean geometries called into question the long-held belief that mathematical axioms are uniquely self-evident. These developments, among others,
brought about a foundational crisis in mathematics at the end of the 1800s that prompted
philosophers and mathematicians to search for a way to ground and unify an increasingly
abstract and voluminous discipline. While a variety of popular foundational theories have
been defended, all of them are controversial in some way and there is no consensus; thus, over
a century later, the foundational crisis still lacks definitive resolution. What seems clear is
that mathematical results such as the non-Euclidean geometries and Gödel’s incompleteness
theorems have shown that mathematics is less certain and absolute than was once believed.
If mathematics and metaphor are not antipodal then the question remains of how they
are related to each other. This dissertation argues that important connexions exist between
mathematics and metaphor, and that exploring and developing these connexions improves
our understanding of both topics. On the one hand, metaphor and analogy seem to comprise
a fundamental mode of human reasoning. This is particularly evident when one considers
that we frequently learn by understanding the unknown in terms of the known. Insofar
as metaphorical reasoning is basic and ubiquitous, one expects it would play some role in
mathematics. Conversely, the modeling capabilities of mathematics are justifiably renowned;
it thus seems reasonable that one could, to at least some extent, mathematically model
metaphor. Both of these approaches are considered below, though more emphasis is placed
3
on the former for several reasons. First, as mentioned above, the original motivation for this
project was an observation that metaphor and analogy are frequently used in mathematics
but the contribution they make generally goes unrecognized. Second, the idea that metaphor
could play an essential role in mathematics remains controversial and unpopular, whereas
significant — and increasing — amounts of effort and resources are allocated to developing
mathematical models of communication and language. There is thus more need for people
to defend and develop the former than the latter. Third, developing a robust mathematical
model of metaphor seems too massive an undertaking to constitute a suitable doctoral project;
in any case, neither my research interests nor my expertise readily lend themselves to such
modeling.
This research project is fundamentally interdisciplinary. While the questions being asked
are primarily philosophical in nature, the project also has a clear mathematical aspect. Additionally, perhaps because metaphors are most apparent and most applicable in mathematical
development, mathematics educators constitute a significant proportion of those sympathetic
to this project. Certainly, the most immediate applications of my research seem to be educational. The fifth chapter of the dissertation employs a somewhat radical interdisciplinary
methodology insofar as mathematical concepts are utilized in non-mathematical ways. Even if
we set aside the mathematical aspects of the project, the subgoal of understanding metaphor
is itself an interdisciplinary undertaking. To various extents, the dissertation draws upon
resources and evidence from cognitive science, linguistics, semiotics, biology, anthropology,
and other related disciplines; however, it should be noted that this document is not intended
as a work in any of these fields. A brief overview of the dissertation will help bring this all
together.
Chapter 2 provides an historical overview of the traditional understanding of metaphor.
The ancient history of the word “metaphor” is traced from its pre-Socratic origins through to
Aristotle, who provides the first explicit account of metaphor as a figure of speech. Aristotle’s
writings form the basis of the traditional view of metaphor developed by science-obsessed
early modern and modern philosophers who took it to be an unnecessary linguistic distraction with no place in science or philosophy. While most scholars of the time endorsed this
traditional viewpoint and its emphasis on literal truth, a small minority of poets and philoso4
phers disputed it, arguing instead that metaphor is an essential device for creative expression
and allows for the expansion and development of language. The twentieth century brought
increased interest in explaining metaphor and, consequently, a variety of novel philosophical theories that remain popular today. Though these theories continue in the literal-truth
tradition, they also tend to be more sympathetic to the creative minority. Ultimately, all of
the theories discussed in this chapter are found to be flawed. The purpose of this synoptical
chapter is to provide motivation and context for the rest of the dissertation by summarizing the most popular philosophical theories of metaphor and examining their strengths and
weaknesses.
Chapter 3 examines the idea that metaphor may be fundamentally conceptual rather than
merely linguistic. The deficiencies observed in chapter 2 have led to increased dissatisfaction with strictly linguistic theories and, consequently, generated support for the conceptual
approach to metaphor. The core of this chapter is an examination of what is arguably the
foremost theory of metaphor-as-conceptual. The conceptual metaphor theory developed by
linguist George Lakoff and his cadre of collaborators — notably Mark Johnson, Mark Turner,
and Rafael Núñez — is the result of an extensive research project spanning three decades and
myriad publications. While the Lakovian theory incorporates a significant amount of empirical research and overcomes many of the problems with linguistic approaches to metaphor, it
nevertheless remains controversial. Because every theory of conceptual metaphor must rely
on an understanding of what a concept is, I begin by introducing Jesse Prinz’s seven desiderata for theories of concepts, thereby providing evaluative structure to the chapter. A few
popular theories of concepts are subjected to Prinz’s criteria; exposing their weaknesses provides further motivation for the alternative Lakovian approach. A relatively detailed synopsis
of Lakoff’s conceptual metaphor theory follows, distilled from a more extensive selection of
publications than most other summaries of his work. An assortment of published criticisms
of Lakoff are addressed; while demonstrating that conceptual metaphor theory satisfies the
majority of Prinz’s desiderata constitutes an adequate defense against most of these criticisms, a few are more telling and suggest areas where the theory could use improvement.
The chapter concludes by briefly noting some promising directions for development, notably
incorporating the work of Cornelia Müller. While this chapter does focus nearly exclusively
5
on Lakovian conceptual metaphor theory, it is not strongly committed to that particular
theory but only to the core idea that many metaphors are conceptual in nature.
With the necessary understanding of metaphor finally established, chapter 4 argues that
metaphor plays an important role in mathematics. Motivation for pursuing alternative,
metaphor-dependent theories of mathematics is derived from considering the serious flaws
of the traditional camps in the philosophy of mathematics; of particular note is a pervasive,
mutual inability to accommodate the strengths of rival positions. The bulk of chapter 4 is
devoted to two leading theories of mathematics-as-metaphor. Lakoff and Núñez’s embodied
mathematics is an offshoot of Lakovian conceptual metaphor theory, an ambitious attempt to
map the constitutive conceptual metaphors underlying mathematical development and practice. Embodied mathematics has been widely criticized by mathematicians and philosophers
alike; however, many of these criticisms fail insofar as they are rooted in equivocation because they depend upon a traditional linguistic understanding of metaphor. Stephen Yablo’s
mathematical figuralism takes a different approach, integrating metaphor with a variety of
mathematical fictionalism. Imre Lakatos’s mathematical quasi-empiricism and Brian Rotman’s semiotic approach are also briefly discussed. The chapter concludes with the suggestion
that a theory of mathematics grounded in an understanding of metaphor-as-conceptual will
have the significant advantage of being able to integrate aspects from both traditional and
contemporary competing theories.
Chapter 5 turns the tables by considering the extent to which metaphor can be understood
mathematically. Scholars have been attempting to rigorously model language and concepts
since at least the time of Leibniz, and the advent of the internet has intensified interest in
such research. However, this chapter does not aim to make a direct contribution to this
project, nor even to provide an overview of the progress that has been made. Without
intending to devalue such rigorous modeling projects, I argue in favour of the non-rigorous
(that is, metaphorical) application of mathematical concepts in understanding metaphor
itself. I conclude chapter 5 by presenting some novel creative mathematical metaphors that I
have found useful in my quest to better understand metaphor. This approach makes chapter
5 the most interdisciplinary chapter of the dissertation. Finally, a brief concluding chapter
indicates some directions for future research.
6
Chapter 2
A Philosophical History of Metaphor
It is a mistake, then, to think of linguistic usage as literalistic in its main body
and metaphorical in its trimming. Metaphor, or something like it, governs both
the growth of language and our acquisition of it.1
— W.V.O. Quine
The purpose of this chapter is to provide a suitable context for the discussion of metaphor
in subsequent chapters by providing some preliminary answers to the question “what is
metaphor?”. Every theory of metaphor must situate itself with respect to traditional and
popular understandings of metaphor by either accounting for them or else explaining why it
need not. Ideally, this chapter would contain a discussion of every way metaphor has been
used or spoken about, both contemporaneously and throughout its long history, in every
human language and culture. For obvious reasons this is not possible; I must be selective
and present a summary of representative fragments of this rich history. One limitation of
scope I can admit explicitly and immediately: I will focus exclusively on metaphor in the
Western tradition.2 While most of the history and theories I present in this chapter have been
given thorough coverage elsewhere in the philosophical and linguistic literature, I believe it
is a worthwhile endeavour to distill the various sources into a single text. Moreover, though
similar compilations exist, none have approached the question of metaphor specifically with
mathematics in mind; this novel perspective will illuminate certain features of the tradition
that are often disregarded, or at least unemphasized.
1
W.V.O. Quine, “A Postscript on Metaphor,” On Metaphor, Ed. Sheldon Sacks (Chicago: U Chicago P,
1979), 160.
2
It could be extremely rewarding to research the origin of the concept of metaphor in other languages and
cultures distant from our own tradition. In particular, it would be worth investigating whether the concept
of metaphor arose independently in other cultures prior to their contact with the Greeks. The results of
such research could have a significant impact on our understanding of metaphor. Such an undertaking would
deviate too far from my current areas of expertise and is thus beyond the scope of this dissertation.
7
It would be useful to have a common preliminary working definition of metaphor to
use as an origin. It need not be perfect or final, it need only correspond to a core set of
rudimentary thoughts about what metaphor is. I recall first learning about metaphor in
elementary school language arts lessons, having been taught a definition something like the
following: “A metaphor is a comparison between two things that does not use the words
‘like’ or ‘as’.” This common schoolbook definition can be seen as a simplified version of one
found in the Oxford English Dictionary (OED):
metaphor, n.
1. A figure of speech in which a name or descriptive word or phrase is transferred
to an object or action different from, but analogous to, that to which it is literally
applicable; an instance of this, a metaphorical expression. Cf. METONYMY n.,
SIMILE n.
2. Something regarded as representative or suggestive of something else, esp. as
a material emblem of an abstract quality, condition, notion, etc.; a symbol, a
token. Freq. with for, of.3
This definition will suffice as a starting point. It manages, however, to also introduce a
difficulty that will plague the investigation: the word “metaphor” has several related but
inconsistent senses. In the OED definition above, the first sense limits “metaphor” to certain
instances of figurative language whereas the second sense is much broader, encompassing all
language insofar as it is representational, and certainly extending to artistic works, etc.; the
second sense can account for metaphors of the first type but the converse relation does not
hold.
Each of the above senses seem to capture some of what is meant by “metaphor” without
being exhaustive; the question thus arises how to best develop and refine a philosophical
understanding of metaphor. Considering a few explicit examples would not do much to
further our understanding at this point; such a process could not resolve the above difficulty,
as good examples for both senses can be found.4 This is not to say that examples are of no
use, or that this chapter will be devoid of them, only that examples alone cannot accomplish
everything desired. Another way I could refine our understanding of metaphor is by following
up on the associated terms suggested by the OED: considering concepts close to
3
METAPHOR
“metaphor, n.,” OED Online, Jun. 2008, Oxford UP, 2 Feb. 2009.
Indeed, the OED is constructed based on word usage evidence, and representative quotations are provided
as part of each entry to demonstrate and justify each listed sense of the word.
4
8
can bring some of its boundary into sharper focus.5 This approach is one of the avenues I will
pursue in chapter 3. However, I think that the best way to begin is to explore the etymology
and history of the word “metaphor.” Oftentimes, knowing the origin of a word will help in
understanding its modern use, though it would be a mistake to assume that etymology is or
must be the arbiter of contemporary meaning.
2.1
Ancient Greece
The word “metaphor” derives from the ancient Greek word µǫτ αφoρά. This word combines
the prefix µǫτ α with the word φoρά to mean “a carrying across” or, less strictly, “a transfer.”6
The noun µǫτ αφoρά is a cognate of the verb µǫτ αφǫ́ρo, a word that occurs more frequently
in the ancient Greek texts.7 Though he does not use the word himself, both the prefix
and root verb of µǫτ αφǫ́ρo are found already in Homer, suggesting that they could have
been conjoined in speech centuries before the eldest surviving written instances.8 The verb
µǫτ αφǫ́ρo remains in use in modern Greek; it can be found, for instance, on the side of
moving vans that can be hired to transfer one’s possessions across town.9 Surprisingly, while
the ancients used this verb to speak of the transfer of duties between individuals and the
transfer of funds in accounts, I could not find a record of any of them using this word to
speak of the transfer of physical objects.10 The use of µǫτ αφoρά that is of primary interest
here concerns linguistic transfers; despite the preceding comment, one might observe that
the conception of linguistic metaphor could itself be seen as metaphoric insofar as it does not
involve any physical carrying.
There is one other noteworthy sense of the word µǫτ αφoρά. It is one of the few surviving
uses of the noun form from ancient Greece. The lengthiest known fragment of an obscure
5
The convention of referring to concepts, conceptual domains, conceptual metaphors, and other cognitive
mechanisms using small capital letters is standard in the literature on conceptual metaphor, and is adopted
throughout this document.
6
“metaphor, n.” OED Online. It is worth noting that the English word “transfer” comes from the Latin
transfero, which is a direct translation of µǫτ αφoρά (“transfer, v.” OED Online).
7
George Henry Liddell and Robert Scott, A Greek-English Lexicon, 9th ed. (Oxford: Clarendon, 1940).
8
Liddell and Scott.
9
Guy Deutscher, The Unfolding of Language (New York: Metropolitan, 2005), 117.
10
Liddell and Scott.
9
poet of the third century BCE named Nicomachus portrays a cook speaking to his employer
on the subtleties of his art. The cook tells the man that a “fully trained” cook must have
mastered a large number of arts before studying cooking, including astronomy, medicine,
and — significantly — geometry. When describing the application of each of these arts in
cooking, the cook uses the term µǫτ αφoρά: the skills are “carried over” into cooking. That
is, when astronomy is used to predict the tides in aid of selecting prime seafood or geometry
is used to optimize the layout of the kitchen, the application of the skill within cooking is a
metaphor insofar as it is a transference.11 This is interesting because it constitutes another
abstract variety of transference, a transfer of skills rather than of physical objects or of words
or meanings; it suggests that, in some sense, the application of mathematical skills might
be seen as involving a kind of metaphor, a view that provides motivation for the position
defended in chapter 4. I now finally turn my attention to linguistic metaphor in ancient
Greece.
Instances of linguistic metaphor are found in the most ancient Greek writings; that is, in
Homer. Both the Iliad and the Odyssey offer up many examples of simile and metaphor.12
Myriad examples of metaphor abound in ancient poetry, and one not need look further than
the writings of Heraclitus to see they figure in the writings of philosophers as well.13 The
linguistic phenomenon known as µǫτ αφoρά was clearly available to be named and conceptualized for at least several centuries before the earliest surviving use of this noun. Many
historical accounts of metaphor start with Aristotle, with good reason — he was the first
to give an explicit philosophical account of metaphor. I shall examine Aristotle’s writings
presently. However, it is known that Aristotle was not responsible for coining the term
µǫτ αφoρά; my investigation thus begins a few years earlier with the man responsible for the
11
Athenaeus, The Learned Banqueters, Ed. and trans. S. Douglas Olson, Vol. 3 (Cambridge: Harvard UP,
2008), 290.
12
John Kirby, “Aristotle on Metaphor”, American Journal of Philology 118 (1997): 521. More on simile
in the discussion of Aristotelian metaphor below, and later in chapter 3. For now, recall the schoolbook
definition: A simile is a comparison between one thing and another that does use either “like” or “as.”
13
For example, see Fragment 52: “Time is a child playing a game of draughts” (Freeman 28). Unfortunately,
my knowledge of the ancient Greek language is limited; thus, I must depend on published translations
throughout this chapter. This introduces a familiar swarm of philosophical questions concerning translation.
Not only will I disregard these questions as being more or less inconsequential to the task at hand, I will
occasionally draw upon multiple translations of a single work for the sake of clarity, thereby strengthening
my arguments.
10
earliest extant usage of the noun µǫτ αφoρά: Isocrates.14
Isocrates (436–338 BCE) was an Athenian rhetorician of some repute; his school was the
main competition for Plato’s Academy.15 His Evagoras is an eulogistic oration addressed to
King Nicocles of Salamis, son of Evagoras; based on this fact, it is assumed to have been
written circa 374 BCE, the year of Evagoras’ death.16 Near the beginning of this eulogy,
Isocrates explains why eulogies are typically written in verse rather than in the prose he
intends to adopt:
To the poets are granted numerous ornaments [of language], for. . . they can express themselves not only in ordinary language, but also by the use of foreign
words (xenois), neologisms (kainois), and metaphors (metaphorais). . . but to writers of prose none of such [resources] are permitted: they must strictly use both
words and ideas [of a certain category:] (1) of words, only those that are in the
[ordinary] language of the polis; (2) of ideas, only those that are closely relevant
to the matter at hand.17
Several important points can be gleaned from this passage. According to Isocrates, there
are two kinds of language: everyday, discursive language and ornamented, poetical language.
Metaphor is one of the kinds of linguistic ornament that distinguishes poetical language from
everyday language. Thus, language is deviant insofar as it contains metaphors and the use
of metaphor is forbidden to everyone except poets. Unfortunately, this passage does not
explain what Isocrates meant by “metaphor,” though it does say something about what
it is not; it is not until Aristotle that a definition of metaphor is provided. John Kirby
suggests that Isocrates may have had a very different notion of metaphor from Aristotle, one
that only refers to the extended poetical similes found in Homer: “it would explain why he
restricts metaphora to verse while at the same time using, in his own prose, what modern
thinkers would call metaphor.”18 Whether Kirby’s suggestion is correct or not, it is easy to
see that Isocrates is dismissive of “metaphor,” whatever the term means to him. Though
Isocrates said relatively little about metaphor and is rarely mentioned in modern philosophy,
his views seem to have had an influence on the tradition, as we shall see. There is one other
14
It is plausible that Isocrates’ legendary rhetorician forebears Corax and Tisias may have written about
metaphor; however, none of their works survive (Hinks 62).
15
Isocrates, Cyprian Orations, Ed. Edward Forster (Oxford: Clarendon, 1912), 9–11.
16
Isocrates 21.
17
Translated in Kirby 523–4. Interspersed parenthetical Greek words removed to enhance readability.
18
Kirby 526. For one example of an extended Homeric simile see Iliad 4.274–82.
11
pre-Aristotelian philosopher that warrants attention due to his profound influence on later
philosophers of metaphor.
Plato (429–347 BCE) did not even use the word µǫτ αφoρά in his writing, let alone provide us with a theory of the phenomenon. However, a few select passages from his dialogues
suggest what some of his thoughts on the matter might have been. These passages are certainly worth considering if for no other reason than their significant influence over subsequent
philosophers. It is relevant to my particular interests that Plato’s theories make a significant
contribution to the development of the traditional academic spectrum, where mathematics
and the imitative arts are located on opposite ends. This antithetical relationship is best
illustrated in the Republic and is grounded in the theory of Forms. Plato’s theory of Forms
holds that everything from physical objects to spatio-temporal instantiations of abstract notions like “justice” are approximations of perfect Forms that transcend the physical realm.19
In Book X of the Republic, Socrates argues that because a painting of a bed is not a bed,
and a bed is not the Form of bed, the works of poets and other imitative artists are three
removes from the truth.20 Because “all poetical imitations are ruinous to the understanding
of the hearers,”21 Plato’s Socrates would either drastically censor all poets or exile them from
his Republic: “we must remain firm in our conviction that hymns to the gods and praises
of famous men are the only poetry which ought to be admitted into our State”
22
On the
other hand, Plato reveres mathematics as one of the best kinds of human knowledge, the
most accurate concrete approximation of the Forms that the human mind can know: “the
knowledge at which geometry aims is knowledge of the eternal, and not of aught perishing
and transient. . . geometry will draw the soul towards truth, and create the spirit of philosophy.”23 Elsewhere, Plato directly situates mathematics in opposition to poetry, claiming
that “measuring and numbering and weighing” help fend off the bewitching falsehoods of the
imitative arts.24 Combining the above results with Plato’s unwavering pursuit of the truth
19
There is certainly more to this theory than can be summed up in a single sentence, but this gloss shall
suffice for my purposes.
20
Plato, Republic, Trans. Benjamin Jowett, The Internet Classics Archive, 597e.
21
Plato, Republic 595b.
22
Republic 607a.
23
Republic 527b.
24
Republic 602d.
12
and Isocrates’ idea that metaphor is one of the characteristic devices of poetry generates
a few important implications. First, the resultant diametric opposition between poetry and
mathematics suggests one explanation why relatively few scholars have looked for connexions
between the two. Secondly, and more importantly, if metaphor is exclusive to poetry and
poetry is far removed from the truth, then one might deduce that Plato held metaphor to
be generally disreputable and worthy of suspicion; certainly, many later thinkers incorporate
such a suspicion of metaphor into their own philosophy, as we shall see. However, there are
good reasons to believe this does not accurately reflect Plato’s position.
The second inference cannot hold because it seems quite clear that Plato did not accept
the Isocratic idea that metaphor is exclusive to poetry; after all, Plato’s own writing is rife
with metaphor. A particularly relevant passage can be found in his Phaedrus:
To tell what [the soul] really is would be a matter for an utterly superhuman and
long discourse, but it is within human power to describe it briefly in a figure; let
us therefore speak in that way. We will liken the soul to the composite nature of
a pair of winged horses and a charioteer. Now the horses and charioteers of the
gods are all good and of good descent, but those of other races are mixed; and first
the charioteer of the human soul drives a pair, and secondly one of the horses is
noble and of noble breed, but the other quite the opposite in breed and character.
Therefore in our case the driving is necessarily difficult and troublesome.25
This excerpt shows, not only did Plato use metaphor, he also held that some truths were
best expressed using metaphor.26 Indeed, Plato was denounced by several ancient critics
specifically for his use of metaphor.27 Does juxtaposing this revelation with the passages
from the Republic expose Plato as a hypocrite? Not at all. Plato clearly realizes that
metaphor is not exclusive to poetry; it is also used in philosophy at the very least. His
attitude toward metaphor is better understood as respectful caution than outright suspicion:
he recognizes metaphor as a powerful tool which, on the one hand, can be used to convey the
most important and difficult truths but, on the other hand, can obfuscate even the simplest
and most basic facts.
25
Plato, Phaedrus, Trans. Harold N. Fowler, Plato in Twelve Volumes (Cambridge: Harvard UP, 1925),
246a–b.
26
Some readers might object that the above passage would be better classified as simile; I believe this
distinction is inconsequential to the point I am making. Indeed, even if one accepts Kirby’s suggestion
regarding Isocratic metaphor — that is, that the term only applies to extended similes such as those found
in Homer — this passage retains its evidential strength.
27
Kirby 530.
13
Thus far, inferences about Plato’s view of metaphor have been made based on his discussion of mimetic poetry as well as on observations of his own abundant use of metaphor.
One other feature of his writing deserves mention. As has already been noted, Plato never
used the noun µǫτ αφoρά in his dialogues. He did, however, use the verb µǫτ αφǫ́ρo in several
places, though never to refer to a transfer of words or meanings.28 His most interesting use
of the word occurs in the Timaeus. In the early pages of the dialogue, Socrates gives a
condensed summary of his ideal State, and then reminds Timaeus, Hermocrates, and Critias
that he had set them the task of depicting this ideal city at war.29 In response to this challenge, Critias famously relates the story of Atlantis: the legendary kingdom once attacked
the Hellenic people, but the Atlanteans were defeated by the Athenians and lost their island
to the ocean depths. Critias then makes the suggestion that the three satisfy Socrates’ request in the following way: “The city and citizens, which you yesterday described to us in
fiction, we will now transfer (µǫτ ǫνǫγκóντ ǫς) to the world of reality. It shall be the ancient
city of Athens, and we will suppose that the citizens whom you imagined, were our veritable
ancestors.”30 The transfer here is of an abstract or fictional entity into an actual scenario.31
This use of the verb µǫτ αφǫ́ρo seems similar to Nicomachus’ use of µǫτ αφoρά, and both are
relevant to my project, as shall become apparent in chapter 4.32
Aristotle (384-322 BCE) is the first scholar to give an explicit account of metaphor. The
bulk of his account is split between the Poetics and Book III of the Rhetoric. In the Poetics,
Aristotle gives the following definition: “Metaphor consists in giving the thing a name that
belongs to something else; the transference being either from genus to species, or from species
to genus, or from species to species, or on grounds of analogy.”33 One important point arises
28
This is not strictly true; in his Critias, Plato uses the word µǫτ αφǫ́ρo to refer to translation between
languages (Kirby 528). This usage foreshadows some ideas that will come up in chapter 4, but at present
seems distantly removed enough from the schoolbook definition to be merely noted.
29
Interestingly, Socrates describes his desire to hear their reply by metaphorically comparing it to the desire
one might have to breathe animating life into a static painting of animals.
30
Plato, Timaeus, Trans. Benjamin Jowett, The Internet Classics Archive, 26c–d.
31
The four conversation partners take the Atlantis story as fact (Plato, Timaeus 20d).
32
Anticipating the discussion of conceptual metaphor in chapter 3, Nicomachus’ and Plato’s underlying conceptual metaphors seem to be COOKING IS GEOMETRY and ANCIENT ATHENS IS THE SOCRATIC IDEAL CITY
respectively.
33
Aristotle, Poetics, Trans. I. Bywater, The Complete Works of Aristotle, Ed. Jonathan Barnes (Princeton:
Princeton UP, 1984), 1457b6–7. The word “epiphora” is used in this definition, and this makes it close to
tautological. Fortunately, one can flesh out this definition by considering other passages. Moreover, following
14
immediately from this definition: for Aristotle, metaphor is primarily a transfer of names or
words, not of meanings or aught else. He follows this definition with illustrative examples
of each of these four types. It has been noted that, in modern terminology, most examples
of the first three kinds would be classified as cases of metonymy or synecdoche rather than
metaphor.34 Moreover, it seems that Aristotle’s second example (“Truly has Odysseus done
ten thousand deeds of worth”) could also be seen as a case of hyperbole.35 It is clear that
the connexions between metaphor, metonymy, and synecdoche must be considered in greater
detail; I will postpone this task to chapter 3. Instances of the fourth type of metaphor, those
based in proportional analogy (“As old age is to life so is evening to day”) seem to be more
typically metaphorical according to the schoolbook definition.36 These analogical metaphors
have several noteworthy features.
Aristotle explains that his fourth type of metaphor is symmetrical: “the proportional
metaphor must always apply reciprocally to either of its co-ordinate terms.”37 For example,
one may either describe evening as the “old age of the day” or describe old age as “the sunset
of life.”38 However, Aristotle also says that proportional metaphors can be used to name the
unnamed; indeed, this is one of the most significant properties of metaphor. For example,
when a new kind of beetle with pointy protuberances coming out of its head is discovered, the
entomologist may note “as antlers are to a stag so Xs are to this beetle” and metaphorically
refer to the protuberances as the antlers of the beetle. This variety of proportional metaphor
where one of the terms is unnamed seems to be fundamentally asymmetrical in that no name
exists to be transferred reciprocally. Considerations of symmetry similar to these will play a
role in any viable theory of metaphor.
Bickerton, Leezenberg notes that Bywater’s translation is erroneous: the phrases “the thing” and “belongs
to something” have no analogues in the original Greek. These additions arguably illegitimately attribute
several presuppositions regarding language to Aristotle that have skewed the interpretations of scholars over
the last century (Leezenberg 36n).
34
Christof Rapp, “Aristotle’s Rhetoric”, Stanford Encyclopedia of Philosophy.
35
Poetics 1457b11–12; qtd. in Kirby 533. Elsewhere, Aristotle notes “Successful hyperboles are also
metaphors.” (Rhetoric, Trans. Roberts 1413a20). His argument for this claim is somewhat unclear, perhaps
he is simply agreeing with my observation.
36
Aristotle, Poetics 1457b22–23. Aristotle notes elsewhere that “Of the four kinds of metaphor, the most
popular are those based on proportion” (Rhetoric, Trans. Freese 1411a1).
37
Rhetoric 1407a14–16.
38
Aristotle, Poetics 1457b16–24. Note also that the first and second types of metaphor are reciprocal to
each other and that the third is self-reciprocal.
15
The ancient Greek for simile is ǫικôν, a word that was also used to refer to visual resemblances and is thus the ancestor of the English word “icon.”39 Aristotle explicitly explains
how metaphor (µǫτ αφoρά) and simile (ǫικôν) are connected: “the simile. . . is a metaphor,
differing from it only in the way it is put; and just because it is longer it is less attractive.”40
Aristotle continues his discussion of simile a few pages later, noting that “Successful similes. . . always involve two relations like the proportional metaphor. . . a simile succeeds best
when it is a converted metaphor.”41 Thus, the connexion between simile and proportional
metaphor is made explicit. It is noteworthy that Aristotle considers simile to be a subspecies
of metaphor; Cicero, Quintilian, and much of the ensuing tradition hold precisely the opposite view, that all metaphors are elliptical similes.42 Note that Aristotle’s claim that all
similes are metaphors does not logically entail that all metaphors are similes. The OED definition of simile varies very little from the ancient understanding: “A comparison of one thing
with another, esp. as an ornament in poetry or rhetoric.”43 I will return to the relationship
between simile and metaphor in chapter 3.
Aristotle differs from those who follow in his tradition by holding simile to be a species
of metaphor. Another divergence from this tradition also constitutes a divergence from his
predecessor Isocrates. Whereas Isocrates holds that metaphor is limited to poetry, Aristotle
claims the following: “these two classes of terms, the proper or regular and the metaphorical
— these and no others — are used by everybody in conversation.”44 While Aristotle does
note that some figures are exclusive to poetry, it is clear that metaphor is not among them:
it is used in prose and in everyday speech.45 Not only does Aristotle see metaphor as more
widespread than Isocrates did, he says nothing to suggest he would have it any other way;
in fact, he enthusiastically remarks that “the greatest thing by far [for a poet] is to be a
master of metaphor. It is the one thing that cannot be learnt from others; and it is also
39
Plato used the word ǫικôν with both of these meanings at various points in his corpus, but these passages
are not very relevant to my purposes; I omit them for the sake of brevity. For more details, see Marsh McCall,
Ancient Rhetorical Theories of Simile and Comparison, Chapter 1.
40
Rhetoric, Trans. Roberts 1410b16–18.
41
Rhetoric, Trans. Roberts 1412b3–4.
42
Mark Johnson, “Introduction,” Philosophical Perspectives on Metaphor, Ed. Mark Johnson (Minneapolis:
U of Minnesota P, 1981), 8.
43
“simile, n.,” The Oxford English Dictionary, 2nd ed., 1989, OED Online, Oxford UP, 18 Feb. 2009.
44
Rhetoric, Trans. Roberts 1404b34–36.
45
Poetics, 1459a10–14.
16
a sign of genius, since a good metaphor implies an intuitive perception of the similarity
in dissimilars.”46 This quote shows that metaphor is not just a matter of technique, but
involves a type of creativity or insight that cannot be taught. It also reinforces the idea that,
for Aristotle, the metaphoric transfer of words is based upon similarity; this is an assumption
that some authors have recently started to question.47 The two quotes in this section prompt
an important question: if metaphor is so widespread and important, what purposes does it
serve?
I have already discussed one important role metaphor plays: the metaphorical naming of
unnamed phenomena and entities, a process sometimes called catachresis. In discussing the
further uses of metaphor, it is important to note the following: “It should be observed that
each kind of rhetoric has its own appropriate style. The style of written prose is not that of
spoken oratory, nor are those of political and forensic speaking the same.”48 Thus, for each
different mode of linguistic communication, metaphor may play a slightly different role in
the corresponding rhetorical style. Aristotle provides the following general account:
The excellence of diction is for it to be at once clear (σαφη̃) and not mean(µὴ
τ απǫιν ὴν).49 The clearest indeed is that made up of the ordinary words for things,
but it is mean. . . On the other hand the diction becomes distinguished and nonprosaic by the use of unfamiliar terms, i.e. strange words, metaphors, lengthened
forms, and everything that deviates from the ordinary modes of speech. But a
whole statement in such terms will be. . . a riddle if made up of metaphors. . . A
certain admixture, accordingly, of unfamiliar terms is necessary.50
Thus, a second general purpose of metaphor is to accentuate truths by making them more
distinct and appealing. The trick, as is often the case with Aristotle, is to aim for the perfect
balance between the extremes; in this case, to seek out the perfect combination of ordinary
words and figures of speech that constitutes an equilibrium between unsophisticated clarity
and unclear sophistication.
46
Aristotle, Poetics 1459a5–8. According to the OED, the word “genius” comes from the Greek γίγνǫσθαι
by way of Latin and has connotations of creation. It is used here as a translation of the Greek word ǫυφυίας
which, according to the Liddell-Scott Greek-English Lexicon, means natural goodness of shape, parts, mind,
or disposition (Liddell and Scott).
47
See, for example, Johnson, “Introduction” 6. However, it seems to me that the phrase “similarity in
dissimilars” leaves some room for interpretation. I will return to this in chapter 3.
48
Aristotle, Rhetoric, Trans. Roberts 1413a3–5.
49
The word “mean” has enough meanings (!) in English to cause some confusion here. To help clarify,
note that other sources translate this word as “commonplace” or “low” or “pedestrian” (Liddell and Scott).
50
Poetics, Trans. Bywater 1458a18–32.
17
Elsewhere, Aristotle says that “Metaphor. . . gives style clearness, charm, and distinction
as nothing else can.”51 This quote suggests, contrary to the passage in the previous paragraph, that some metaphors can simultaneously increase both distinction and clarity. One
hypothesis is that metaphors may sometimes increase clarity by means of improved efficiency
of communication; an apt, brief metaphor can be richly meaningful.52 The idea of aptness
— that instances of metaphor can be better or worse — is one of the key issues in the study
of metaphor. In the Rhetoric, Aristotle argues
the fallacious argument of the sophist Bryson [says] that there is no such thing
as foul language, because in whatever words you put a given thing your meaning
is the same. This is untrue. One term may describe a thing more truly than
another, may be more like it, and set it more intimately before our eyes. Besides,
two different words will represent a thing in two different lights; so on this ground
also one term must be held fairer or fouler than another.53
That Aristotle argues this point is not surprising; indeed, one of the fundamental justificatory
assumptions of rhetoric is that there is more than one way to express a given truth. Metaphor
is certainly one important mechanism that allows for this variation in expression. While
people ought to accept or reject ideas primarily, or even solely, on the basis of truth, the
particular wording one uses in communicating a truth may play an important supporting
role in convincing an audience.54 If metaphor can improve communication, one might expect
that it can also hinder it. Aristotle goes on to explain that there are several common ways for
metaphors to fail to be appropriate: they may be too tragic, too farfetched, too ridiculous.55
This brief glimpse into Aristotle’s views on what makes for an apt metaphor provides a
starting point for a contemporary understanding.
This discussion of the second purpose of metaphor originated with a crucial distinction.
Recall that Aristotle presents an opposition between unfamiliar terms and ordinary words;
this distinction is identical to the one made by Isocrates, though Aristotle is friendlier to
51
Rhetoric, Trans. Roberts 1405a8–9.
To slightly alter a cliché to fit my purposes, a figure [of speech] is worth a thousand words.
53
Rhetoric, Trans. Roberts 1405b11–15.
54
For example, by using language that is appealing to a specific audience, a speaker may attract their
undivided attention and make them more receptive to the truth being communicated through humour, pity,
fear, etc. Note that though these techniques of language have important positive uses, they may also be used
fallaciously to convince an audience of falsehoods — see, for example, the fallacy of mob appeal.
55
Rhetoric, Trans. Roberts 1406b6.
52
18
unfamiliar words. While Aristotle has much to say on metaphor in the Rhetoric and the
Poetics, he says almost nothing about ordinary words. In the Poetics, he says “By the
ordinary word, I mean that in general use in a country.”56 However, this does not explain
very much as it says nothing explicit about the ultimate grounding of this general usage.57
Aristotle’s contrast between ordinary and unfamiliar words bears a strong resemblance to the
distinction between literal and figurative language utilized by scholars in theorizing about
metaphor for at least the last four hundred years.58 As “metaphor” is sometimes defined as
being synonymous with “non-literal,” it is somewhat surprising to discover that relatively
little explicit consideration has been given to the question “what is literality?” by those who
study metaphor. The second purpose of metaphor — to contribute attractive and enlivening
sophistication to language — raises several deep and important issues that require attention,
particularly those related to aptness and literality.
A third purpose of metaphor is that it can be used to teach. This usage is related to
catachresis in that every thing a person encounters was once new and nameless to them.
That metaphor has a didactic usage is revealed by Aristotle in the following passage:
we all naturally find it agreeable to get hold of new ideas easily: words express
ideas, and therefore those words are the most agreeable that enable us to get
hold of new ideas. Now strange words simply puzzle us; ordinary words convey
only what we know already; it is from metaphor that we can best get hold of
something fresh.59
Whenever a person encounters a new idea, they tend to compare it to things and ideas they
are familiar with, looking for similarities. This process can provide the foundation for an
analogical metaphor that allows one to speak and reason about the novel idea. Under this
56
Aristotle, Poetics 1457b3–4.
It is possible that Aristotle says something more about ordinary speech elsewhere, in some part of his
corpus I am less familiar with. I suspect that, like many later scholars who discussed metaphor, Aristotle
has at least partially taken for granted that the idea of ordinary language is obvious and needs little further
definition. Whether they are privy to explication unknown to me or have made inferences from his broader
theory, the editors of Liddell and Scott’s Greek-English Lexicon define Aristotle’s term for ordinary words,
κύριoν, as “the real or actual, hence current, ordinary, name of a thing” (Liddell and Scott).
58
The English word “literal” derives from the Latin word litteralis, an adjective meaning roughly “pertaining
to [alphabetic] letters.” In English, the relevant definition for my purposes is “[pertaining] to the etymological
or the relatively primary sense of a word, or to the sense expressed by the actual wording of a passage, as
distinguished from any metaphorical or merely suggested meaning.” This meaning derives from an earlier
usage describing a mode of Biblical interpretation that contrasts with the mystical or allegorical (“literal,
adj. and n.” OED Online).
59
Rhetoric, Trans. Roberts 1410b9–13.
57
19
view, it becomes clear that one way a teacher may induce learning is by drawing analogies
between ideas familiar to their students and unfamiliar material from the curriculum. Because
a person may fail to be taught to craft good metaphors, those who are predisposed towards
metaphor mastery have an advantage both in teaching and in learning as they have the
potential to create powerful new metaphors as the need arises.60 Elsewhere, Aristotle calls
into question the value of metaphor and other stylistic devices in teaching: “in every system
of instruction there is some slight necessity to pay attention to style; for it does make a
difference, for the purpose of making a thing clear, to speak in this or that manner; still,
the difference is not so very great, but all these things are mere outward show for pleasing
the hearer.”61 The above two quotations seem difficult to reconcile under the assumption
that metaphor is solely stylistic: they agree that metaphor has some place in instruction and
learning, but differ drastically on the importance of that role.
The only way I can see to resolve this tension is by explaining why the second quotation
does not apply to metaphor. One option is to consider the possibility that metaphor has nonstylistic uses; though Aristotle’s discussion of metaphor is dominated by considerations of its
sophisticating purpose, the brief discussion of catachresis makes this hypothesis promising,
and I will give it further consideration below. Another possibility is that Aristotle is using
the word “style” as a metaphor of the first kind; that is, he is transferring the word for a
genus, “style,” to one of its species, “delivery.” Thus, by saying that style does not play
a big role in instruction, Aristotle might only mean that teachers need not have sonorous
voices and vocal skills on par with actors. This interpretation seems somewhat spurious but
is supported by the fact that the second quotation is taken from a passage which primarily
focuses on delivery; unfortunately, my knowledge of Greek is not sufficient to further test
this hypothesis. This incongruity seems of little consequence given that most of Aristotle’s
writing on metaphor supports the first quotation, and the second quote does not even contain
the word “metaphor”; why then am I giving it thorough consideration? The reason is that
the sentence immediately following the second quotation contains one of the only references
to mathematics in the Rhetoric: “Nobody uses fine language when teaching geometry.”62
60
Aristotle, Poetics 1459a5–8.
Rhetoric, Trans. Freese 1404a9–13.
62
Aristotle, Rhetoric, Trans. Roberts 1404a13. The only other reference to mathematics in the Rhetoric
61
20
Insofar as mathematics can be considered a mode of communication, it too must have its own
associated rhetorical style. This quote suggests that Aristotle may have held that geometry
should be conducted entirely in ordinary language, that metaphors play no role in geometrical
education.63 One might question whether Aristotle can be considered an expert regarding
mathematics instruction; while he taught a great number of disciplines, none of his surviving
works are specifically about mathematics. As one of the motivating intuitions of my project
is that mathematical practice necessarily involves metaphor, the apparent incongruity in
Aristotle’s writing on the use of metaphor in teaching discussed in the last two paragraphs
is of great interest.
There are several reasons why I have given a relatively lengthy treatment to Aristotle’s
theory of metaphor. First, it is justifiably considered to be the original account of metaphor;
this fact alone makes it interesting and worthy of discussion. Furthermore, the above summary of his account shows it to be both descriptive — as it provides a definition for metaphor
and situates the concept with respect to related concepts — and prescriptive — insofar as it
explains some of the appropriate and inappropriate uses of metaphor. This comprehensive
scope is rare among theories of metaphor and provides further incentive to situate contemporary theories in relation to Aristotle. In particular, Aristotle motivates discussion of the
symmetry features of metaphor, of the relation of metaphor to simile and analogy, of what
makes an apt metaphor, of what constitutes literality and how it is related to metaphor, and
of the use of metaphor in didactics.
Many academics have chosen to build their account of metaphor in a fairly narrow tradition based upon Aristotle so having a solid comprehension of his theory will assist with
understanding subsequent authors. Indeed, there is so little deviation from the aforementioned tradition for thousands of years that consideration of a few representative authors
will be sufficiently illustrative; as Umberto Eco notes, “of the thousands and thousands of
pages written about the metaphor, few add anything of substance to the first two or three
I am aware of is of less consequence at this juncture, but I quote it here for the sake of completeness:
“mathematical discourses depict no character; they have nothing to do with choice, for they represent nobody
as pursuing any end” (1417a19–20).
63
It is worth observing that another potential avenue of resolution is to consider whether Aristotle meaningfully distinguished between teaching and learning, holding that the role of metaphor is slight in the former
but great in the latter.
21
fundamental concepts stated by Aristotle.”64 Although Eco might not have had the same
fundamentals in mind, Mark Johnson’s suggestion that “the future of metaphor is prefigured
in terms of these three basic components [of Aristotle’s theory]: (i) focus on single words that
are (ii) deviations from literal language, to produce a change of meaning that is (iii) based
on similarities between things”65 certainly seems to be consistent with Eco’s claim. Johnson
goes on to suggest that the following definition of metaphor unifies the dominant tradition:
“A metaphor is an elliptical simile useful for stylistic, rhetorical, and didactic purposes, but
which can be translated into a literal paraphrase without any loss of cognitive content.”66
While Johnson’s first quote is accurate, there seem to be some fundamental divergences
between Aristotle’s theory and the tradition described by the latter quote.
A final reason for reviewing Aristotle’s theory in detail is to suggest that his theory should
not be located within the tradition he is often credited with originating. There are several
reasons for thinking this. First, as noted above, Aristotle believed that simile was a species of
metaphor; Cicero was among the first influential thinkers to invert this relation.67 One does
not have to be the father of categorical logic to appreciate that {similes} ⊂ {metaphors} and
{metaphors} ⊂ {similes} have very different repercussions. Second, nowhere does Aristotle
say that metaphors are eliminable; rather, his suggestion that one must seek a mean between
ordinary and unfamiliar language suggests the opposite. This second divergence from the
tradition seems to depend in part on the first. Similarly, the third is related to the second:
many scholars in the tradition not only think that metaphor is eliminable, they believe
it ought to be eliminated. This view corresponds more closely to Isocrates than to Plato
or Aristotle. To be sure, both Plato and Aristotle are occasionally unfriendly to poetic
language, unfamiliar terms, and metaphor in their writing insofar as they can be taken to
excess and used for sophistry. However, they also both expound the virtues of the moderate
use of similes, metaphors, and figurative language in various types of linguistic discourse —
including the philosophical. The similarities and differences between Aristotle’s theory and
those of his philosophical descendants will become more apparent after further consideration
64
Umberto
Johnson,
66
Johnson,
67
Johnson,
65
Eco, Semiotics and the Philosophy of Language (Bloomington: Indiana UP, 1984), 88.
“Introduction” 6.
“Introduction” 4.
“Introduction” 8.
22
of a few representative members of the tradition.
The history of the concept of metaphor is a tree. Its roots are Aristotle and his contemporaries. A straight trunk with almost no branches stretches upward out of these roots
until the twentieth century, where the first seriously acknowledged branches diverge. By the
1970s, the crown of the tree develops in earnest, with many twigs protruding from several
branches. Thus, as the investigation continues through history, there is an increasing pressure to generalize and simplify rather than consider each individual twig. This is unfortunate
but necessary, as a thorough chronicling is beyond the scope of this dissertation and, indeed,
probably beyond the scope of any single paper. The primary goal of this chapter is to provide
context for discussion of non-traditional contemporary theories of metaphor; considering the
major trends in metaphor theory along with a few key representative figures will be adequate
for this purpose. Given the way theories develop, a more or less chronological approach
remains appropriate.
2.2
Early Modern and Modern Philosophy
For many centuries, theories of metaphor mostly consisted of gradually diminishing echoes
of Aristotle’s discussion. As already noted, the most notable deviation from Aristotle is the
inversion of the genus to species relationship of metaphor and simile, an idea supported by
Cicero in his De Oratore and in most subsequent writings on the subject. This reversal
seems strongly correlated to metaphor’s diminishing status in the writings of rhetoricians
and theologians throughout late antiquity and the middle ages. While the role of metaphor
in philosophy was at first simply deemphasized, eventually its noted positive contribution to
non-poetic modes of language was entirely forgotten, eclipsed by condemnation on grounds of
being a utensil of sophistry. These sentiments were taken up by many of the science-obsessed
philosophers of the early modern era.68
Hobbes (1588–1679) is an ideal representative of this tradition: he provides an extreme
stance on metaphor which, upon brief reflection, reveals him as a hypocrite. While it is
the theory of the commonwealth presented in the Leviathan that is most often discussed,
68
Johnson, “Introduction” 9–11.
23
the book begins with a scientifically themed discussion of human beings and their mental
capacities. In the fourth chapter “Of Speech,” Hobbes roughly defines metaphor as the use
of words “in other sense than that they are ordained for” and ranks it as one of the four
abuses of language on the grounds of its deceptive character.69 Furthermore, Hobbes says
that metaphor is the sixth cause of absurdity in reasoning: “though it be lawful to say, for
example, in common speech the way goes or leads hither or thither, the proverb says this or
that, whereas ways cannot go nor proverbs speak, yet in reckoning and seeking of truth such
speeches are not to be admitted.”70 For Hobbes, metaphor is the non-literal and has no place
in science, philosophy, or any serious, truth-seeking enterprise. This view is far less detailed
than that of Aristotle. Hobbes and Aristotle agree that metaphor is non-literal, though
Aristotle counts many other species of unfamiliar language. However, they clearly disagree
in their prescriptions: whereas Aristotle believes metaphor use is good in moderation, Hobbes
would eliminate metaphor completely, at least within academia.
Given Hobbes’ explicit unwavering rejection of metaphor, one might expect a scarcity of
non-literal language in his writing. This is obviously not the case. The very title of his magnum opus, Leviathan, betrays a pervasive foundational metaphor:
HUMAN.
THE STATE IS A POWERFUL
The first paragraph of Hobbes’ Introduction introduces the metaphor
LIVING THINGS
MACHINES ARE
in detail: “what is the heart, but a spring; and the nerves, but so many strings;
and the joints, but so many wheels.”71 Perhaps most condemning, the following quotation
comprises the conclusion of the fifth chapter of Leviathan, “Of Reason, and Science”:
To conclude, the light of human minds is perspicuous words, but by exact definitions first snuffed and purged from ambiguity; reason is the pace; increase of
science, the way; and the benefit of mankind, the end. And, on the contrary,
metaphors, and senseless and ambiguous words, are like ignes fatui [a fool’s fire],
reasoning upon them is wandering amongst innumerable absurdities; and their
end, contention and sedition, or contempt.72
69
Thomas Hobbes, Leviathan, Ed. Edwin Curley (Indianapolis: Hackett, 1994), I.iv.4. The word “ordained”
potentially has problematic religious connotations. It suggests that literality might depend upon divine
decree. Hobbes’ presentation of the Babel myth in I.iv.2 suggests he may be simply using the word “ordained”
to mean “proper” here. I note this merely as a possible explanation of literality, albeit one that is unacceptable
for my purposes.
70
Hobbes I.v.14.
71
Hobbes 1.
72
Hobbes I.v.20. Bracketed comment included in text. Ignes fatui are also known as will-o’-the-wisps,
ghostly lights known for luring people into bogs.
24
This passage beautifully portrays metaphor as the antithesis of the goods of reason and science; however, Hobbes uses metaphor in his very disparagement of metaphor! This apparent
hypocrisy provides evidence in favour of the rejection of extreme anti-metaphor stances.
This conclusion may have interesting implications for mathematics.
Like Descartes,
Hobbes attempts to make his project rigorous by making impositions on it analogous to
those found in mathematical reasoning, and these are largely responsible for his rejection of
metaphor.73 In his discussion on speech, Hobbes claims that “without words there is no possibility of reckoning of numbers, much less of magnitudes.”74 He notes that people without
language might have some extremely basic arithmetic and geometric skills, but would lack
the ability to abstract from particular objects and figures.75 Thus, if one accepts that mathematics depends on words, and that attempting to eliminate all metaphor from discourse
in order to introduce mathematical rigor is a fool’s errand, then one might conclude that
metaphor has some role to play in mathematical reasoning. This argument involves a lot of
speculation, hypotheticals, and some controversial assumptions, but it gestures towards the
position that will be defended in chapter 4.
Hobbes’ view of metaphor provides an illustration of what Johnson calls “the literaltruth paradigm,” a position characterized by a belief in the primacy of literal language,
with metaphor being an eliminable deviation from the literal.76 Hobbes is not the sole
advocate of this paradigm; indeed, the majority of theories of metaphor are variations on
this theme. While not all those who fall under the literal-truth paradigm are as hostile
to metaphor as Hobbes, many of his philosophical contemporaries had similar sentiments.
George Berkeley (1685–1753), for example, simply proclaims “a philosopher should abstain
from metaphor.”77 John Locke (1632–1704) is more vicious in his condemnation, stating that
all figurative language is “for nothing else but to insinuate wrong ideas, move the passions,
and thereby mislead the judgment” and is therefore “wholly to be avoided,” concluding
that “where truth and knowledge are concerned, [figures of speech] cannot but be thought
73
For more on Descartes’ desire to use the rigorous deductions of axiomatic geometry as the model for all
inquiry, see his Discourse on Method (1637).
74
Hobbes I.iv.10.
75
Hobbes I.iv.9.
76
Johnson, “Introduction” 12.
77
From Of Motion, quoted in Johnson, “Introduction” 12.
25
a great fault, either of the language or the person that makes use of them.”78 Sometime
in the eighteenth century philosophical discussion of metaphor seems to have fallen out of
vogue, possibly due to stagnation; the few theories that were published tended to simply be
uninspired restatements of the literal-truth paradigm expressing a general animosity towards
metaphor.79 It would take the renewed interest in the philosophy of language at the end of
the nineteenth century to bring metaphor under serious philosophical scrutiny once again.
Above, I have provided a sketch of the primary traditional view of metaphor from its roots
in the ancient Greeks through nearly two-and-a-half millennia of development and dormancy.
However, not everybody subscribes to the literal-truth paradigm; considering some of these
alternative positions will add crucial breadth to this chapter. The non-traditional views of
metaphor considered in this chapter exhibit enough commonalities to justify the application of
a classificatory term. As these viewpoints are unanimous in recognizing creative imagination,
and metaphor in particular, as fundamentally important, I shall say they are representative
of the creative-imagination paradigm.80 A few general remarks are in order. First, note that
simply being friendly towards metaphor is not a sufficient condition for membership in this
class. Second, it is not entirely clear when this branch diverged from the main tradition.
Its origins can be seen in certain passages in Aristotle.81 However, the particular views
considered below originate in the eighteenth and nineteenth centuries. I suspect that there
have been proponents of the creative-imagination paradigm throughout history whose beliefs
have been lost for various reasons.82
78
John Locke, Essay Concerning Human Understanding, Ed. Peter H. Nidditch (Oxford: Clarendon, 1975),
III.x.34.
79
Johnson, “Introduction” 13.
80
The words “imagination” and “paradigm” should be understood in a loose and common rather than technical way here: no philosophically substantial claims were intended by the particular choice of terminology.
81
For example, Aristotle on catachrestical metaphor in the Poetics, 1457b24–29.
82
Some hypotheses:
1. Because the creative-imagination paradigm tends to lend support to theories regarding the origin and
development of language that could be seen as heretical, proponents of this paradigm were often not
vocal in supporting their beliefs until a point in history when it was safer to do so.
2. Anyone who did write in support of the primacy of metaphor would be more likely to utilize poetical
or ostentatious metaphor in that writing, thereby making it more likely to be rejected for publication
by proponents of the literal-truth paradigm.
3. There seems to be a correlation between the aforementioned decrease of publications admonishing the
use of metaphor and the increase in publications supporting the creative-imagination paradigm.
26
Where the literal-truth paradigm has the scientifically minded philosopher as an archetypal advocate, the creative-imagination paradigm has the Romantic poet. One might expect
this, as poets rely on metaphor as one of their fundamental professional tools. To some
extent, this claim is based upon anecdotal evidence as most poets opt to write poetry rather
than treatises on metaphor, just as most practicing mathematicians would prefer to do mathematics than contemplate its foundations.83 However, a few poets have left records of their
beliefs on these matters. Percy Bysshe Shelley’s (1792–1822) A Defence of Poetry is a prime
example of such a record.84 In this paper, Shelley presents the following view of metaphor:
[The language of poets] is vitally metaphorical; that is, it marks the before unapprehended relations of things, and perpetuates their apprehension, until words,
which represent them, become, through time, signs for portions and classes of
thought, instead of pictures of integral thoughts; and then, if no new poets should
arise to create afresh the associations which have been thus disorganised, language
will be dead to all the nobler purposes of human intercourse.85
This passage claims that, far from being an ornamental deviation from the literal, metaphor
actually precedes the literal. It also suggests that without poets to inject life into language,
it would die and wither, becoming useful only for the most base tasks. This is the first time
the notion of dead language has appeared in this chapter, but it becomes a pervasive theme
in later writing on metaphor. This synopsis of Shelley’s view of metaphor is substantiated
further down the page:
In the infancy of society every author is necessarily a poet, because language itself
is poetry. . . Every original language near to its source is in itself the chaos of a
cyclic poem: the copiousness of lexicography and the distinctions of grammar
are the works of a later age, and are merely the catalogue and the form of the
creations of Poetry.86
83
One anecdotal source is provided by Johnson: “It was Romantic artists and poets, rather than philosophers, who preserved and celebrated the notion of creative imagination” (“Introduction” 14–5). However,
Johnson provides no explicit examples to back up this claim.
84
Shelley’s paper is a response to his friend Thomas Love Peacock’s The Four Ages of Poetry. In this work,
Peacock situates himself in the literal-truth paradigm: “Feeling and passion are best painted in, and roused
by, ornamental and figurative language; but the reason and the understanding are best addressed in the
simplest and most unvarnished phrase,”(Peacock 9) and hence “While the historian and the philosopher are
advancing in, and accelerating, the progress of knowledge, the poet is wallowing in the rubbish of departed
ignorance, and raking up the ashes of dead savages to find gewgaws and rattles for the grown babies of the
age.”(Peacock 15). One wonders whether his criticisms of poetry using poetical language are intentional and
ironic, or unintentional and hypocritical.
85
Percy Bysshe Shelley, “A Defence of Poetry,” Peacock’s Four Ages of Poetry, Shelley’s Defence of Poetry,
Browning’s Essay on Shelley, Ed. H.F.B. Brett-Smith (Oxford: Blackwell, 1929), 26.
86
Shelley 26.
27
Those who subscribe to the creative-imagination paradigm also tend to espouse comparable
views about the development of language; namely, that definitions and dictionaries only follow
where poets have already blazed metaphoric trails.87 The very origin of the ongoing project of
representing and reasoning about the world in language must lie in the human poetic faculty,
the creative imagination. The essayist Thomas Carlyle summarized this developmental stance
most aptly and succinctly: “The coldest word was once a glowing new metaphor.”88
Although most philosophers of this period favoured the literal-truth paradigm, a few were
vocal proponents of some version of the creative-imagination position. For example, JeanJacques Rousseau (1712–1778) can be seen to support the creative-imagination paradigm in
his Essay on the Origin of Languages: “As man’s first motives for speaking were of the passions, his first expressions were tropes. Figurative language was the first to be born. . . At first
only poetry was spoken; there was no hint of reasoning until much later.”89 Contemporary
advocates of a creative-imagination stance often recognize Giambattista Vico (1668–1744) as
the father of their philosophical tradition. A rhetorician and philosopher, Vico is perhaps
best known for his magnum opus The New Science, an ambitious work that seeks to establish “the principles of humanity,” to give an account of the development and functioning of
typically non-quantifiable human practices and institutions. His method in this undertaking
constitutes a “new science” insofar as it attempts to overcome deficiencies he perceived in
the then-dominant Cartesian hypothetico-deductive method by acknowledging the necessary
contribution of human individuals and societies to all things. His approach is to form a
synthesis of philosophy and philology — thereby vitalizing the truths of the former using the
certainties of the latter — and to use this “new critical art” to examine human experience.90
Vico summarizes the end of his undertaking as follows: “Thus our Science comes to be at
once a history of the ideas, the customs, the deeds of mankind. From these three we shall
87
The term “poet” is sometimes used more generally to describe anyone who invokes creative imagination,
and not only those who do so in writing poetry (in the narrow sense). Shelley certainly was friendly to such a
general interpretation: “Poetry, in a general sense, may be defined to be ‘the expression of the Imagination’ ”
(Shelley 23). The justification for this is typically etymological: “poetry” derives from the Greek word πoιǫι̂ν,
a verb meaning to create, make, compose, prepare, write, produce, etc. (Liddell and Scott).
88
Thomas Carlyle, Past and Present, Ed. Chris R. Vanden Bossche (Berkeley: U of California P, 2005),
130.
89
Quoted in Johnson, “Introduction” 15.
90
Timothy Costelloe, “Giambattista Vico,” Stanford Encyclopedia of Philosophy, 11 Mar. 2009.
28
derive the principles of the history of human nature, which we shall show to be the principles
of universal history, which principles it seems hitherto to have lacked.”91
In the first pages of The New Science, Vico examines the history of language as centrally
important in the history of humanity. Vico’s assumptions lead him to conclude that “the
principle of the origins both of languages and of letters lies in the fact that the early gentile
people, by a demonstrated necessity of nature, were poets who spoke in poetic characters.
This discovery. . . is the master key of this [new] Science.92 That is, the contribution that
humans made to the origins of language must have ultimately been a work of the creative
imagination. In particular, according to Vico, metaphor played an important role in the
origins of language and therefore in the history of humanity:
The most luminous and therefore the most necessary and frequent [of the first
tropes] is metaphor. It is most praised when it gives sense and passion to insensate things, in accordance with the metaphysics above discussed, by which the
first poets attributed to bodies the being of animate substances, with capacities
measured by their own, namely sense and passion, and in this way made fables
of them. Thus every metaphor so formed is a fable in brief. This gives a basis for
judging the time when metaphors made their appearance in the languages. All
the metaphors conveyed by likenesses taken from bodies to signify the operations
of abstract minds must date from times when philosophies were taking shape.
The proof of this is that in every language the terms needed for the refined arts
and recondite sciences are of rustic origin.93
For Vico, metaphor was the characteristic trope of the second of the three ages of the world,
the Heroic age. Metaphor allowed poets like Homer to extend the simple representative
hieroglyphs of the first age beyond their natural immediate referents. It thus allowed language
to address entities and activities of a less corporeal nature: “vulgar Latin. . . has formed almost
all its words by metaphors drawn from natural objects according to their natural properties
or sensible effects. And in general metaphor makes up the great body of the language
among all nations.”94 There are a couple of important points to take from Vico. First, his
91
Giambattista Vico, The New Science of Giambattista Vico, Trans. Thomas Bergin and Max Fisch (Garden
City, New York: Doubleday Anchor, 1961), §368, 73. It is important to note the motivation and nature of
Vico’s book because of their drastic departure from the Cartesian project to make all thought approach the
standard set by deductive geometric proof based upon clear and distinct axioms; the Cartesian “geometric”
approach, the literal-truth paradigm, and the notion that mathematics is devoid of metaphor are all intimately
interconnected.
92
Vico, §34, 5.
93
Vico, §404, 87–8.
94
Vico, §444, 104.
29
injection of etymology and history into philosophy helps vindicate the approach taken in this
chapter. More generally, his approach suggests that a theory of metaphor ought to include
some considerations of the development of language. And second, Vico is possibly the first
philosopher to note that a large proportion of words in contemporary use involve metaphor
in their etymological roots. This is a refreshing revelation after the hypocrisy of Hobbes.
It must be noted, however, that Vico’s theory suggests that all words eventually find their
ground in the signs based in natural relation and resemblance that comprised the language
of the first age; the extent to which Vico considered these first signs to be metaphorical is
unclear. One final pre-twentieth century philosopher warrants consideration for deviating
from the literal-truth paradigm even more radically than Vico.
Friedrich Nietzsche (1844–1900) presents a view of metaphor in one of his earliest works,
On Truth and Lies in an Extra-Moral Sense.95 In this manuscript, Nietzsche questions why
humans desire truth, especially given how deeply deception is connected to biology; for
example, he explains how deception forms a part of several successful survival strategies.
He concludes that the drive for truth originates in the fact that humans are social animals
and therefore need a peace agreement to prevent descent into a Hobbesian state of nature.
However, this origin has significant implications for Nietzsche’s understanding of truth:
What then is truth? a mobile army of metaphors, metonyms, anthropomorphisms, in short, a sum of human relations which were poetically and rhetorically
heightened, transferred, and adorned, and after long use seem solid, canonical,
and binding to a nation. Truths are illusions about which it has been forgotten
that they are illusions, worn-out metaphors without sensory impact, coins which
have lost their image and now can be used only as metal, and no longer as coins.96
Nietzsche’s view is strongly antithetical to the literal-truth paradigm, in that it posits that all
words and signs are necessarily metaphorical because things-in-themselves, and hence capitalT Truths, are transcendentally beyond our reach. This perspective is similar to the Vichean
in emphasizing metaphor and its constitutive role in language, though Nietzsche’s stance is
definitively more extreme: he claims that “every concept originates through the equation
95
Though Nietzsche apparently later rejected the Romanticism of his early writings, this paper is worthy
of consideration as part of this context chapter due to its unique outlook on metaphor.
96
Friedrich Nietzsche, “On Truth and Lying in an Extra-Moral Sense,” Friedrich Nietzsche on Rhetoric
and Language, Ed. and trans. Sander L. Gilman, Carole Blair, and David J. Parent (New York: Oxford,
1989), 250.
30
of the dissimilar,” and concludes that even the uptake and integration of neural stimuli
constitutes metaphor.97 Interestingly, Nietzsche further exposes his Kantian assumptions by
claiming that the intuitions of space and time (as discussed at length in the Critique of Pure
Reason) and their mathematical regularity are presupposed in the creation of all metaphors,
and thus transitively in concepts, the well-worn residues of metaphors. Nietzsche seems to
suggest, then, that at least some of mathematics might not be metaphorical insofar as it is
part of, or emergent from, the regularity-providing intuitions that are a precondition for the
fabrication of metaphors. Indeed, mathematics seems to be part of this regularity. Despite
some confusing Kantian-flavoured remarks about mathematics and a few incorrect or obsolete
biological assumptions, Nietzsche’s radical claims about metaphor are ahead of their time
insofar as they seem to anticipate several important features of the Lakovian theory of a
century later that is the focus of chapter 3.
The relatively small motley of poets, artists, and philosophers who were the early supporters of the creative-imagination paradigm made some important contributions to the
philosophy of metaphor that are reflected in many of the theories of the twentieth century.
Foremost among these contributions is the idea that human creativity plays an essential role
in the origin and development of language; this supposition entails that metaphor is not
merely an afterthought — a disposable garnish on the enchilada of literality — but rather
precedes and makes an etymological contribution to many now-common words. This line
of thought reaches its apex in Nietzsche’s approach which presciently considers the development of language as part of the phylogenetic development of the human species. It is
in the developmentally and biologically oriented writings of the creative-imagination theorists that the notion of dead metaphor first arises in this historical summary; though the
phrase is not explicitly invoked, the idea is clearly present. At first glance, the metaphor of
“dead metaphor” seems an easy way to understand the developmental ideas of the creativeimagination theorists: novel metaphoric phrases gain popularity and, through time and use,
lose their metaphoricity. The idea of dead metaphor is not consistent with extreme versions
of the literal-truth paradigm (as it acknowledges metaphor as more than merely derivative),
but does seem to fit with more moderate claims regarding the primacy of literal language.
97
Nietzsche 249.
31
Because of its apparent aptness and its consistency with a wide variety of theories, many
subsequent authors have used the idea of “dead metaphor” without providing significant
explication. Though the creative-imagination theorists provided some important ideas to
the philosophy of metaphor, many people shy away from adopting the paradigm wholesale,
probably because of a worry that it weakens the notion of truth. As such, the literal-truth
paradigm remained dominant into the twentieth century.
2.3
Twentieth Century
The powerful logical positivist movement that occurred in philosophy during the first half of
the twentieth century exhibited extreme devotion to the literal-truth paradigm. Members of
this movement emphasized empirical verifiability of statements as crucial to meaningfulness,
and this caused them to be suspicious or outright dismissive of statements that failed to fully
live up to this criterion. Famously, logical positivists were highly critical of the statements
of theology and ethics, as well as positive universal claims.98 In his “Lecture on Ethics,”
Wittgenstein shows that logical positivism was also dismissive of metaphor:
if I can describe a fact by means of a simile I must also be able to drop the simile
and to describe the facts without it. Now in [ethical and religious language] as
soon as we try to drop the simile and state the facts which stand behind it, we
find that there are no such facts. And so, what at first appeared to be simile now
seems to be mere nonsense.99
Though this passage is explicitly examining the use of simile in theology and ethics, it betrays
certain presuppositions and biases regarding simile and, arguably therefore, metaphor. What
appears to be a simile must either be merely a garnished version of a factual statement that
can be reached through paraphrase, or else be a pseudo-simile, an empty lump of garnish
deceptively masquerading as a paraphrasable simile, that is, “mere nonsense,” devoid of
meaning. Insofar as this passage is representative of the logical positivist movement, it
exposes them as being as negative regarding metaphor as the early modern empiricists.
Thus, though the philosophy of language gained new momentum in the early twentieth
98
John Skorupski, “Later Empiricism and Logical Positivism,” The Oxford Handbook of Philosophy of
Mathematics and Logic, Ed. Stewart Shapiro (New York: Oxford UP, 2005), 65–9.
99
Ludwig Wittgenstein, “Lecture on Ethics,” The Philosophical Review 74.1 (1965): 10.
32
century, most philosophers remained orthodox advocates of the literal-truth paradigm and
did not advance new theories of metaphor. I am aware of only one significant counterexample
to this trend.100
I.A. Richards (1893–1979) was a philosopher and a rhetorician who was influential within
the study of English literature. Richards’ 1936 book The Philosophy of Rhetoric contains
a chapter on metaphor, the thesis of which is that some of the major assumptions of the
literal-truth paradigm date back to Aristotle and have prevented the development of a suitable theory of metaphor. Insofar as Richards denies that metaphor is deviant and instead
claims that “[the fact] that metaphor is the omnipresent principle of language can be shown by
mere observation,”101 he reveals himself as an advocate of the creative-imagination paradigm.
This classification is further substantiated by his recognition that Shelley and historians of
language were among the exceptional few who saw past the problematic tradition. There
are two significant ways in which Richards goes beyond his forebears in developing a significant theory of metaphor. First, Richards notes that “when we use a metaphor we have two
thoughts of different things active together and supported by a single word, or phrase, whose
meaning is a resultant of their interaction.”102 A claim rather like this is already made by
Aristotle; what makes Richards’ assessment significant is that he goes on to make the observation that “One of the oddest of the many odd things about the whole topic is that we have
no agreed distinguishing terms for these two halves of a metaphor — in spite of the immense
convenience, almost the necessity, of such terms if we are to make any analyses without confusion.”103 Richards proposes the adoption of some technical jargon to help eliminate some of
the vagueness and ambiguity: the tenor of a metaphor is the underlying idea or subject that
is conveyed by means of the vehicle. Clumsily, the tenor might be described as “the original
idea” while the vehicle is “the borrowed idea.”104 Richards concludes that the adoption of
this terminology serves to circumvent the problematic imposition of imagery terminology,
100
It should be remarked that my present familiarity with philosophical literature of this period in no way
precludes the existence of other relevant papers and authors. As was noted at the beginning of this chapter,
this historical survey is necessarily a selective sampling of the relevant literature, and one of the unfortunate
but necessary constraints on this process is the limits of my knowledge. Please forgive any glaring omissions.
101
I.A. Richards, “Lecture V: Metaphor,” The Philosophy of Rhetoric (Oxford: Oxford UP, 1936), 50.
102
Richards 51.
103
Richards 52.
104
Richards 52.
33
and also allows the insight that “in many of the most important uses of metaphor, the copresence of the vehicle and tenor results in a meaning (to be clearly distinguished from the
tenor) which is not attainable without their interaction.”105 Richards goes on to note that
it would be worthwhile to consider interactions between tenor and vehicle based in relations
other than resemblance, including relations of disparity.106 Thus, Richards’ careful dissection of metaphor gives a deeper insight into what it might be and how it might function.
The second revelation of Richards’ lecture is that “[t]hought is metaphoric, and proceeds
by comparison, and the metaphors of language derive therefrom. To improve the theory
of metaphor we must remember this.”107 The idea that metaphoric sentences are merely
instantiated symptoms of underlying metaphoric thought is a powerful one that is later used
efficaciously by Lakoff. Richards’ account is more provocative than definitively detailed,
but his novel insights nonetheless constitute an important contribution to the philosophy of
metaphor.
The next significant landmark in the history of the philosophy of metaphor is Max
Black’s (1909–1988) seminal 1955 paper “Metaphor.” Though Black was clearly influenced
by Richards in the writing of this paper, it is “Metaphor” and not The Philosophy of Rhetoric
that marks the beginning of a new age in the philosophy of metaphor. There seem to be
three main reasons for this. First, while Richards was a rhetorician and literary theorist
first and foremost, Black was an eminent and respected philosopher of language at the time
“Metaphor” was published.108 This likely resulted in his work having a wider exposure
among philosophers and thus more apt to provoke philosophical responses. Second, the
nearly twenty-year gap between Richards’ and Black’s publications saw some developments
in the philosophy of language (for example, the emergence of ordinary language philosophy)
that seem to have rendered the 1950s a more receptive time for the types of ideas presented
in these papers. And third, Black’s treatment of metaphor is nicely systematic, a paragon instance of a philosophical paper. While Black published other works dealing with the subject
105
Richards 55.
Richards 59–60.
107
Richards 51; emphasis his.
108
A relevant but tangential fact: Max Black possessed a B.A. in mathematics from Cambridge and was
employed as a math lecturer previous to receiving his professorship in the philosophy of language (O’Connor
and Robertson).
106
34
of metaphor, I shall focus on his views presented in “Metaphor.”109
As noted above, one of the key merits of Black’s paper is its systematicity. He opens the
paper by noting the widespread acceptance by philosophers of the commandment “Thou shalt
not commit metaphor” and suggesting this has contributed to a dearth of adequate theories.
Black then suggests a list of questions that any adequate theory of metaphor should provide
answers to:
How do we recognize a case of metaphor?
Are there any criteria for the detection of metaphors?
Can metaphors be translated into literal expressions?
Is metaphor properly regarded as a decoration upon “plain sense”?
What are the relations between metaphor and simile?
In what sense, if any, is a metaphor “creative”?
What is the point of using a metaphor?110
By making these questions explicit, Black provided important structure both for his own
paper and for subsequent authors. In the wake of these questions, Black presents a collection of sentences that he hopes will be unmistakably recognized as examples of metaphor.
Though the sentiment is admirable, this is one of the points where his theory seems weakest.
Whereas Black claims that “The rules of our language determine that some expressions must
count as metaphors” regardless of the “occasions on which the expressions are used,” later
pragmatic theories of metaphor will convincingly argue that the textual context a sentence
is uttered in plays a key role in the interpretation of that sentence, including whether it
is interpreted metaphorically or literally.111 Black’s examples consist only of isolated sentences and therefore are devoid of textual context, leaving them potentially ambiguous.112
One might take this conclusion further and claim that even when extended textual passages
are quoted as examples in philosophical papers on metaphor, the transposition into a new
context necessarily has an impact on interpretation; thus, quoted metaphors are like animals in captivity: one must not assume that their “behaviour” is fully natural. Another
109
For more of Black’s thoughts about metaphor, see his “More About Metaphor” (1979) and Models and
Metaphors (1962).
110
Max Black, “Metaphor,” Proc. of Aristotelian Society 55 (1954-55): 64.
111
Black, “Metaphor” 66.
112
Note that the metaphoricity of Black’s first example seems less compellingly unmistakable when it is
given some context: “The chairman plowed through the discussion. Nine were killed by the madman’s rusty
implement.”
35
potential difficulty surrounds the presentation of explicit examples. Black seems to suggest
elsewhere that the concept of
METAPHOR,
like that of
GAME,
is connected by Wittgensteinian
family resemblance.113 An intuition very much like this one is the primary reason for the
conspicuous absence of examples from this chapter so far. In any case where a complicated
category is being considered, there is always a danger that examples given will be mistakenly
taken as paradigmatically definitive of the category by the reader. That is, whether a robin,
a penguin, or an ostrich is presented as an example of a bird, it is likely to introduce a bias
that skews the reader’s understanding of the category
BIRD.
To be fair, Black has provided
a wide range of examples to help fend off this concern, something that many other authors
fail to do. However, there is a key difference between birds and metaphors that is relevant
here: whereas new birds may be discovered that challenge the understanding of the category
emerging from the presentation of multiple examples, novel metaphors may be explicitly
constructed to serve this function. Thus, the nature of metaphor seems to make it ill suited
for explication through even an extensive array of examples. The bird example was chosen
specifically to both explain the introduction of bias and to illustrate it on a meta-level by
introducing a bias regarding family-resemblance concepts. The reader is urged to remain cognizant of the warnings of this paragraph whenever a list of example metaphors is encountered
in the literature.
Based on his provided examples, Black makes the following observation: “In general,
when we speak of a relatively simple metaphor, we are referring to a sentence or another
expression, in which some words are used metaphorically, while the remainder are used nonmetaphorically. An attempt to construct an entire sentence of words that are used metaphorically results in a proverb, an allegory, or a riddle.”114 He then implements some technical
jargon to reinforce the above observation: the focus of a metaphorical sentence is defined
as the word or phrase being used metaphorically within a non-metaphoric frame. While
this distinction is reminiscent of Richards’, it is important to note that the tenor/vehicle
113
Andrew Ortony, “Metaphor: a multidisciplinary problem,” Metaphor and Thought, Ed. Andrew Ortony,
1st Ed. (New York: Cambridge UP, 1979), 5.
114
Black, “Metaphor” 65.
36
distinction is not identical to the focus/frame distinction.115 Black’s quote also seems to pay
homage to Aristotle’s suggestion that a balance between metaphoric and non-metaphoric
speech is desirable, with an excess of metaphor giving rise to riddles. A couple of additional
general claims about metaphor that Black makes are relevant to this discussion. Black argues that metaphor is not a syntactic phenomenon — the recognition and interpretation of
metaphorical sentences seems to be independent of grammar — but rather has to do with
meanings of sentences. His evidence for this claim comes from his belief that the very same
metaphor can be encoded in multiple languages.116 He concludes that metaphor is therefore
within the domain of semantics, though he acknowledges a pragmatic dimension that “may
be the one most deserving of attention”: the intended gravity of the focus — the “depth”
of the metaphor — seems to be strongly context dependent.117 Unfortunately, Black spends
little time pursuing this observation. Instead, the remainder of the paper is dedicated to the
discussion of three competing views of metaphor.
Black rejects what he labels substitution and comparison views of metaphor before presenting his own interaction view. Substitution views of metaphor hold that every instance of
a metaphorical sentence is a substitute for some equivalent literal expression.118 Such views
have dominated throughout the history of the subject, as has been noted above, and lead
to the condemnation of metaphor as distracting and potentially confusing ornamentation. If
such a view were true, what purposes could metaphor possibly serve? Black suggests two
possibilities: metaphor could be employed catachrestically when no literal equivalent exists
or its use could be merely stylistic.119 Though the nomenclature was not invoked, substitution views were criticized earlier in this chapter for being hypocritical and contradicting
linguistic evidence, and Black likewise rejects them, moving on to consider comparison views.
Any comparison view of metaphor can be seen as a specific variety of substitution view, in
115
That is, while focus and frame are both defined as parts of a sentence, vehicle and tenor seem more akin
to sentence and speaker meaning, respectively.
116
Black, “Metaphor” 66.
117
Black, “Metaphor” 67.
118
Black, “Metaphor” 68.
119
Black notes that “[i]t is the fate of catachresis to disappear when successful” (“Metaphor” 69); that is,
such usages tend to be conventionalized into the lexicon if a gap indeed exists. It is the suggestion that all
metaphors are catachrestic that Donald Davidson objects to when he says “to make a metaphor is to murder
it” (“What Metaphors Mean” 437). One arguably positive stylistic use of metaphor outside of literature is
in euphemism, the use of inoffensive terms to communicate potentially offensive ideas.
37
that “it holds that the metaphorical statement might be replaced by an equivalent literal
comparison”;120 that is, comparison views postulate that metaphors are elliptical similes.
Not only do comparison views have to answer the same criticisms as substitution views (why
bother to express the simile indirectly?), they are subject to a further difficulty; as Black puts
it, they suffer from “a vagueness that borders upon vacuity” because of the unconstrained
breadth of the likeness relation: any thing is like any other thing in various, and usually
numerous, ways.121 Thus, saying that a metaphor is a comparison without significant further
explication of the notion of comparison does little to no explanatory work. Black contends
that his interaction theory overcomes both the redundancy and vacuity defects.
Black introduces his interaction theory with the following gloss: “when we use a metaphor
we have two thoughts of different things active together and supported by a single word,
or phrase, whose meaning is a resultant of their interaction.”122 If the meaning of such
metaphors emerge through such an interaction, then it follows that neither of the two interacting components are redundant and, therefore, eliminable through substitution.123 Another
interesting consequence of the positing of such emergent meanings is that metaphors seem to
be able to create similarities in some cases rather than being necessarily based in pre-existing
similarity.124 The metaphoric interaction is supposed to proceed as follows: the frame of a
metaphor forms a context for the focus that extends its meaning by inviting the audience
to see one thing as another. Black claims that talk about some thing evokes a “system of
associated commonplaces,” a collection of accurate attributions, half-truths, stereotypes, and
120
Black, “Metaphor” 71.
“Metaphor” 71.
122
Black, “Metaphor” 72.
123
Even if one accepts Black’s interaction theory as correct, it seems that some particularly shallow
metaphors could be successfully eliminated through substitution. In his later work, “More About Metaphor,”
Black implements jargon in saying that ineliminable and unparaphrasable metaphors are more emphatic (26).
Towards the end of “Metaphor,” Black puts forward the suggestion that metaphors might be taxonomically
classified as substitution, comparison, or interaction metaphors (78). Meaningfully following this line of
thought further would seem to require the discussion of the pragmatics of metaphor that Black does not
provide.
124
Given the previous paragraph’s remark about the breadth of the likeness relation, this sentence could
use some clarification. Though it might be accurate to say “everything is like everything else in numerous
ways,” perhaps this ought to be qualified by “in potentia.” While some likenesses or resemblances have
been repeatedly recognized and thoroughly conventionalized as similarities, other metaphoric and similitive
sentences seem to require a more substantial creative effort. For example, contrast “Arms are like tentacles”
with “Ravens are like writing desks.” The idea that metaphors are at least sometimes creative in this way is
one I will want to insist on.
121
38
other sentiments regarding that subject that are commonly held within some culture.125 It
is the systems of commonplaces associated with each of the two “subjects” of the metaphor
that interact, the “subsidiary subject” organizing the “primary subject” by emphasizing certain of its aspects and de-emphasizing others. For example, in Black’s example “Man is a
wolf,” “man” is the primary subject and “wolf” is both the subsidiary subject and the focus
of the metaphor. The metaphor invites the audience to see men as wolves via the basic
cultural understanding of wolfhood, and thus serves to extend the meaning of “wolf” to have
a sense that is applicable to “man” while maintaining its original senses.126 Given this brief
sketch of Black’s interaction theory, a few important remarks are required. First, Black’s
view does not explain which precise aspects will be (de-)emphasized or the magnitude of
this (de-)emphasis; as previously noted, further progress in this direction would require some
discussion of the pragmatics of metaphor. A second related point is that in some cases, an
author may contrive a textual context that is meant to augment or override the system of
associated commonplaces: “in a poem, or a piece of sustained prose, the writer can establish a novel pattern of implications for the literal uses of the key expressions, prior to using
them as vehicles for his metaphors.”127 Thirdly, Black contends that interaction metaphors
have a bidirectional symmetry: “If to call a man a wolf is to put him in a special light,
we must not forget that the metaphor makes the wolf seem more human than he otherwise
would.”128 Finally, and perhaps most significantly, Black notes that the interaction view as
he has presented it employs metaphor in its discussion of metaphor (metaphor is a filter,
metaphor is a projection, etc.). While such an employment was hypocritical in Hobbes, it
seems only reasonable in Black’s writing given his claim that “ ‘Metaphor’ is a loose word, at
best, and we must beware of attributing to it stricter rules of usage than are actually found
in practice”;129 indeed, he explicitly states that he has “no quarrel with the use of metaphors
(if they are good ones) in talking about metaphor. But it may be as well to use several, lest
125
Black, “Metaphor” 74.
Black, “Metaphor” 73–4.
127
Black, “Metaphor” 77.
128
Black, “Metaphor” 77.
129
Black, “Metaphor” 66. It is no surprise that Black is comfortable with loose concepts: his early work
on vague sets and language anticipates the advent of fuzzy logic by nearly three decades (O’Connor and
Robertson).
126
39
we are misled by the adventitious charms of our favourites.”130 Hence, interaction metaphors
are powerful, irreplaceable devices that can be used to great effect in organizing and communicating complex and important meanings throughout the various areas of human discourse
— including philosophy. However, one must remain cognizant of the potential dangers of
using such metaphors: they can depend on — and therefore perpetuate — false, harmful
stereotypes, and if an interaction metaphor becomes an exclusively dominant way of seeing
some thing, it can be problematically limiting insofar as such metaphors tend to obfuscate
some important aspects of their primary subject in order to illuminate others.
Black’s interaction view is an elaboration and refinement of ideas found in Richards, and
provocatively defies the traditional views by suggesting that certain strong metaphors are
ineliminable and may be legitimately used in philosophical discourse. The question remains
whether Black’s theory belongs to either of the two paradigms posited in this chapter, as
it seems to fall somewhere between the “Metaphor is eliminable ornamentation” and “All
language is metaphoric” extremes. On the one hand, Black’s theory emphasizes the ineliminability and creativity of a class of metaphors; on the other hand, his theory does seem
to appeal to an underlying notion of literality in its understanding of metaphor at times,
insofar as literality plays some role in the system of associated commonplaces. Perhaps the
best answer to this question is that Black’s theory does not seem to be clearly contained in
either of these paradigms, but rather served to restructure the way metaphor was viewed.
Rather than simply characterizing metaphor as eliminable or essential, the relative deluge of
papers written in Black’s wake tend to focus on identifying or classifying metaphors, or on
explaining how they work and when they are useful, or on considering the cognitive status
of metaphor.131 Thus, the literal-truth versus creative-imagination classification seems ill
equipped to handle the multidimensionality of metaphor studies in the later half of the twentieth century. Fortunately, Black’s paper seems to recommend some alternative classification
schemes. One such scheme is presented explicitly by Black himself: theories of metaphor could
be classified as substitution, comparison, or interaction views. Another possibility would be
to attempt to classify theories based on their answers to the list of questions posed by Black
130
131
Black, “Metaphor” 73.
Johnson, “Introduction” 20.
40
in “Metaphor.” However, it is the distinction between pragmatic and semantic theories that
seems primarily relevant in classifying the theories of metaphor in the period between Black
and Lakoff. Most of the leading theories from this time period are pragmatic, a trend that
seems natural given Black’s provocative suggestion that the pragmatics of metaphor might
be of primary import despite his own semantic stance. However, it is worth spending a little
time considering semantic theories before moving on to the predominant pragmatic trend.
Not all views of metaphor after Black are pragmatic. Though Black is the first author
to identify his position as a semantic one, earlier substitution and comparison views are
also typically semantic; thus, advocates of traditional theories tend to be semantic theorists.
One such was Paul Henle, whose contribution to the philosophy of metaphor was to update
some of the traditional claims about metaphor using C.S. Peirce’s account of symbols and
icons.132 However, in the post-Black period, most semantic theories of metaphor bore a
closer resemblance to his interaction theory than to traditional views. Michiel Leezenberg
includes Black’s interaction theory in a class of views he refers to as descriptivist that also
includes those of Monroe Beardsley and Nelson Goodman.133 Descriptivist views of metaphor
are united by the idea that the interpretation of a metaphor is “guided by the descriptive
information associated with an expression” rather than being a matter of a similarity or
some other relation between the referents of a metaphorical expression.134 Views like those
of Cicero and Henle are classified as referentialist because they satisfy this latter criterion.135
The number of referentialists dwindled shortly after the advent of descriptivist theories and
their criticisms of various tenets of referentialism.136 Similarly, many semantic descriptivists
converted to a pragmatic approach once it had been established as a genuine alternative;
the later writings of Max Black suggest he was among said converts. While clear examples
of both referentialist and descriptivist semantic theories exist, some important authors defy
classification.
132
Johnson, “Introduction” 25.
Leezenberg 10–1.
134
Leezenberg 11.
135
Leezenberg 71. Leezenberg situates this classification scheme orthogonal to the semantic/pragmatic
divide.
136
Though I will not discuss their views here, it should be noted that referentialist viewpoints did not die
out altogether post-Black. Two later referentialists mentioned by Leezenberg are Mooij and Fogelin (74).
133
41
Paul Ricoeur’s La métaphore vive (translated into English as The Rule of Metaphor) is
among the most voluminous works on metaphor to date. Ricoeur’s work repeatedly identifies
itself as semantic, but his suggestion that he will reconcile substitution and interaction by
appealing to hermeneutics makes it somewhat unclear whether his view ought to be classified
as referentialist, descriptivist, or as belonging to some heretofore unnamed third category.137
Due to the scope of Ricoeur’s work and its connexions to the phenomenological tradition, it
is not feasible to summarize it here, even in brief; however, two especially relevant passages
must be mentioned before moving on. First, Ricoeur claims that the most important theme
of his work is that “metaphor is the rhetorical process by which discourse unleashes the
power that certain fictions have to redescribe reality.”138 This alleged connexion between
metaphor and fiction provides motivation for the discussion of mathematical fictionalism in
chapter 4. Second, “the ‘place’ of metaphor, its most intimate and ultimate abode, is. . . the
copula of the verb to be. The metaphorical ‘is’ at once signifies both ‘is not’ and ‘is like.’ ”139
The combination of these two significations imbues the copula with the sense of being-as.140
Although it is not immediately clear how this remark can be true of metaphorical expressions
that do not contain a copula, it is resonant with the standard
TARGET IS SOURCE
notation
for conceptual metaphors used by conceptual metaphor theorists. Having dispensed with
semantic views for the time being, I now consider pragmatic accounts of metaphor.
Whereas semantic views of metaphor hold that there is some kind of alteration or bifurcation of the meanings of some words in every metaphorical expression, pragmatic views
hold word meanings constant and instead suggest metaphorical interpretation occurs apart
from meanings, at the level of the conversational participants. Many supporters of a pragmatic view of metaphor adopt an approach rooted in H.P. Grice’s theory of conversational
implicature.141 In “Logic and Conversation,” Grice makes the observation that there is an
apparent divergence in meaning between the connectives of formal logic and the words in
English typically associated with them; anyone who has taken a logic course is familiar with
137
Paul Ricoeur, The Rule of Metaphor, Trans. Robert Czerny (Toronto: U Toronto P, 1975), 66.
Ricoeur 7.
139
Ricoeur 7.
140
Ricoeur 257.
141
Some notable authors adopting such a viewpoint include Merrie Bergmann, Stephen Levinson, A.P.
Martinich, and, of course, H.P. Grice.
138
42
the difficulties and confusion that can arise from the differences between understandings of
the symbolic formal connective ∨ and the English word “or,” for example.142 Motivated by
the observation that what is communicated often differs from or extends beyond what is explicitly said, Grice jargonizes the verb “implicate” to describe such communicative acts. By
distinguishing between the conventional meanings of an utterance and its implicata, Grice
bypasses the quagmire of controversy and disagreement that comes with the positing of multiple simultaneous divergent meanings, allowing for the construction of a positive theory.
Though there are several varieties of implicature, Grice argues that metaphor is a variety of
non-conventional, conversational implicature.143
There is a cluster of extralinguistic conventions governing conversations insofar as they
are rational cooperative communicative exchanges. Principal among these is the Cooperative
Principle: “Make your conversational contribution such as it is required, at the stage at
which it occurs, by the accepted purpose or direction of the talk exchange in which you
are engaged.”144 This principle works in conjunction with four categories of maxims which
Martinich helpfully integrates into the following supermaxims:
Quality: Do not participate in a speech act unless you satisfy all the conditions
for its nondefective performance.
Quantity: Make your speech act as strong as appropriate but not stronger than
appropriate.
Relation: Make your contribution to the conversation one that ties in with the
general course of the conversation.
Manner: Make your contribution brief, clear, orderly, and unambiguous.145
142
H.P. Grice, “Logic and Conversation,” Syntax and Semantics, Eds. Peter Cole and Jerry L. Morgan (New
York: Academic Press, 1975), 165. “Meaning” here must refer to non-natural meaning, the kind of meaning
that is created by humans and arises through a reciprocal recognition between speaker and audience that
empowers the intention behind an utterance. Talk of intentions and non-natural meanings is conspicuously
absent from “Logic and Conversation.” For more details on non-natural meaning, see Grice’s “Meaning”
(1957).
143
Conventional implicatures are those which arise directly from the conventional meanings of words and
the rules of language. For example, the inference from the utterance “Bob and Doug drank beer” to the
conclusion “Bob drank beer” is a case of conventional implicature. Non-conventional implicatures depend on
institutions or other factors beyond the standard conventions of language (Grice, “Logic and Conversation”
167).
144
Grice, “Logic and Conversation” 167.
145
A.P. Martinich, “A Theory of Fiction,” Philosophy and Literature 25 (2001): 100. Martinich’s formulation
of the supermaxims is superior to Grice’s categories in being more concise and simultaneously more general
insofar as they cover speech acts other than assertions.
43
It is through the expectation that conversation participants are adhering to the Cooperative
Principle and the four supermaxims that conversational implicatures arise. In some cases,
a conversation partner may make a contribution that obviously and ostentatiously fails to
satisfy one of the supermaxims; the obviousness of the failure suggests that the speaker
intended for the failure to occur and also intended for the audience to notice the failure.
Thus, the obviousness of the failure suggests that the speaker is not intending to deceive
and can be assumed to be otherwise conforming to the conversational maxims; therefore, the
audience can assume that the ostentatious failure is relevant and thereby seek out implicata
that make sense of the contribution. When such an ostentatious failure occurs, Grice says
that a maxim has been flouted, and when a flouting is used to generate an implicature, he says
that a maxim has been exploited.146 Paraphrased into Martinich’s terminology, Grice’s claim
is that metaphorical utterances flout the supermaxim of Quality by characteristically being
categorically false and that the implicatures are often grounded in resemblances.147 He also
notes that metaphors are sometimes used ironically and therefore concludes that implicature
is layerable.148 Grice’s discussion of metaphor is rather scant, but A.P. Martinich elaborates
on how metaphors implicate.149
Martinich explicitly notes that his Gricean theory of metaphor is “blatantly pragmatic”
insofar as whether a sentence flouts the supermaxim of Quality usually depends on its context
of use; thus, both the recognition and interpretation of metaphor are strongly context dependent for Martinich.150 The following is a summary of Martinich’s account of how implicature
occurs for a typical metaphor:
1. The speaker utters “Jeanne is a fox.”
2. The hearer recognizes that a) due to context, the utterance is literally false (since
146
Grice, “Logic and Conversation” 170.
Aristotle seems to suggest the rudiments of a mechanism like implicature in the Poetics: “whenever
also a word seems to imply some contradiction, it is necessary to reflect how many ways there may be of
understanding it in the passage in question” (Trans. Bywater 1461a32–34).
148
Grice, “Logic and Conversation” 172.
149
While Martinich’s theory seems mostly consistent with Grice’s, there are a few noteworthy differences:
a) Grice used the less worrisome phrase “conventional meaning” in the types of situations where Martinich
uses the more inflammatory “literal meaning”; b) The few sentences Grice provided about metaphorical
implicature suggest that his understanding might be better classified as a comparison view, while Martinich’s
elaboration seems to be an interaction view.
150
Martinich, “A Theory for Metaphor,” Journal of Literary Semantics 13 (1984): 457.
147
44
Jeanne is a person and hence not a fox); b) such an obvious falsehood must be a case
of flouting the supermaxim of Quality; and, therefore, c) the speaker’s utterance must
have been a making-as-if-to-say intended to communicate by way of implicature.151
3. The hearer compiles a list of the various distinctive characteristics she attributes to
foxes that seem relevant within the overall context (including considerations of the participants in the conversation, the histories of the participants and their shared history,
the culture they live in, the setting of the conversation, the other communicative exchanges in the conversation, etc). Note that perceived salient characteristics need not
be correctly attributable to the thing in question, but merely shared by the speaker and
hearer for the communication to come off properly. Due to context dependence and
the possible detachment from fact, a salience may be born, appealed to, and forgotten
within the span of a single conversation. The idea of salient characteristics bears much
similarity to Black’s “system of associated commonplaces.”
4. Using the original utterance and the list of salient characteristics, the hearer constructs
an argument:
Jeanne is a fox.
A fox is sly, or cunning, or. . .
Therefore, Jeanne is sly, or cunning, or. . .
The conclusion of this argument is the implicatum of the metaphorical utterance.152
Martinich notes that these implicating arguments support an interaction view of metaphor
(the second premise of the syllogism is the locus of interaction), and alleges that its indeterminate disjunctions conflict with the literal paraphraseability supposition of comparison
views. Later in the article, Martinich observes that, rarely, nonstandard metaphors occur
that are literally true; one example might be “Jeanne is an animal.” Such metaphors require
an additional layer of interpretation: by saying something patently obvious and therefore
151
At the beginning of his article, Martinich distinguishes between saying-that and making-as-if-to-say to
eliminate an ambiguity he observes in Grice’s writing; making-as-if-to-say occurs when an utterance is spoken
in order to generate an implicature without committing to the meaning of the utterance. Martinich argues
that Merrie Bergmann’s account of metaphor is defective insofar as it conflates these two notions (“A Theory
for Metaphor” 448).
152
Martinich, “A Theory for Metaphor” 450–2.
45
redundant (humans are animals, after all), they flout the supermaxim of Quantity, which
prompts the audience to suppose they are false, allowing interpretation to proceed as in the
standard case.153 It is worth noting that Gricean theories of metaphor uphold the literaltruth paradigm to the extent that an utterance that flouts the Quality maxim necessarily
deviates from the literal. Considering the less explicitly Gricean theory presented by John
Searle will help bring further definition to the pragmatic understanding of metaphor.154
Searle’s account of metaphor is an offshoot of his general theory of speech acts. For Searle,
“the main problem of metaphor is to explain how speaker meaning and sentence meaning
are different and how they are, nevertheless, related.”155 Searle’s distinction between literal
sentence meaning and speaker’s utterance meaning is somewhat analogous to the Gricean
distinction between what a sentence says and what it implicates, and his view is similarly
pragmatic as it explicitly asserts that “[m]etaphorical meaning is always speaker’s utterance
meaning.”156 For Searle, the first stage in metaphor interpretation is the identification of
metaphor, so he owes an explanation of how metaphorical utterances are distinguished from
both literal utterances and non-literal, non-metaphoric phenomena such as irony and indirect
speech acts.157 Admirably, Searle recognizes that the recognition task requires a characterization of literal utterances and notes that most authors fail to address or even to recognize
this “extremely difficult, complex, and subtle problem.”158 There are three features of literal
utterance that Searle contends are relevant to his account of metaphorical utterance. First,
153
Martinich, “A Theory for Metaphor” 454.
While I believe Searle would agree with me that his theory is not Gricean, Elisabeth Camp and other
authors have used his view as the prime example of a Gricean theory (Camp and Reimer 849). This taxonomic
controversy does not warrant further attention.
155
John Searle, “Metaphor,” Expression and Meaning (Cambridge: Cambridge UP, 1979), 87.
156
Searle, “Metaphor” 77.
157
Recall that Grice pointed out that metaphor may be employed ironically; for example, one may say to
their nemesis “You are the cream in my coffee” (Grice, “Logic and Conversation” 172). It should also be
noted that though Searle limits his discussion of metaphors to the class of speech acts he calls assertions,
one can also find instances of metaphor in each of his other four taxa, as well as among the indirect speech
acts. Searle explains informally that an indirect speech act involves saying one thing and meaning not only
that thing, but something else in addition; the canonical example is explicitly asking “Can you pass the
salt?” to indirectly communicate a request to have the salt passed (“Indirect Speech Acts” 30). Building
on this, an example of an utterance that is an indirect speech act and employs metaphor ironically: at a
dinner party, I might ask one of my dinner companions “Can you pass the nectar of the gods?” in reference
to a particularly awful bottle of wine. Savas Tsohatzidis claims that this counts as evidence against Searle’s
theory (Tsohatzidis 368–9).
158
Searle, “Metaphor” 78.
154
46
in a literal utterance, speaker meaning and sentence meaning are identical. Second, even in
the most straightforwardly literal utterance, a context created by background assumptions
beyond the semantic content is typically required to establish truth conditions for the sentence.159 And third, any account of literal predication must rely upon an understanding of
similarity.160 These three observations suggest that the interpretation of literal utterances
depends only on a knowledge of the rules of language and an awareness of the contexts and
background assumptions that resolve the indexical and referential elements of the sentence,
whereas the interpretation of metaphor must invoke some further principles; in particular,
Searle notes that “[a]n analysis of metaphor must show how similarity and context play a role
in metaphor different from their role in literal utterance.”161 Giving a precise account of what
these principles are and how they allow an audience to arrive at different truth conditions
for an utterance than those determined by its literal meaning is what Searle calls “the hard
problem of the theory of metaphor.”162
Given his discussion of literal language, Searle suggests that the primary strategy for
metaphor detection involves the recognition of some defect arising if the utterance is taken
literally, including obvious falsehood, semantic nonsense, or violations of the rules of speech
acts or of conversational maxims.163 Utterances that are literal failures are also examined as
possible cases of irony (where speaker meaning is the exact opposite of sentence meaning)
or indirect speech acts (where sentence meaning is a non-exhaustive component of speaker
meaning). Once a potential metaphor has been detected, the hearer appeals to a set of
principles to help compute potential alternative meanings. Searle contends that no single
principle will suffice, and suggests a partial list of principles that can be involved when
one thing calls another thing to mind, including being a necessary or contingent property
159
Searle discusses this point in greater detail in his “Literal Meaning,” arguing that there is no such thing
as a null context for sentence interpretation (117).
160
Searle, “Metaphor” 81.
161
Searle, “Metaphor” 93.
162
Searle, “Metaphor” 85.
163
Searle, “Metaphor” 105. He notes that this can not be the sole strategy for metaphor detection given the
existence of what Martinich calls “nonstandard metaphors.” He also notes that another strategy depends
on whether the particular author or speaker under consideration is known for being prone to metaphor use,
such as in the case of a Romantic poet or a Zen master.
47
of, being falsely conventionally related, being similar, through a part-whole relation, etc.164
Finally, the list of possible meanings is restricted to relevant ones by way of some shared
strategies that include considering the textual context of the uttered sentence and comparing
the predicates involved in the sentence and possible speaker meanings. While it seems Searle
would hold Martinich’s account to be problematically simple, Martinich contends Searle’s
view is problematically loose and his principles are vacuous; their theories seem practically
identical in all other respects.165
Searle’s theory is more extensive than the above summary suggests, and there are several
further relevant points that deserve consideration. Searle’s Principle of Expressibility says
“whatever can be meant can be said,” and to the extent he holds this to be true he must
give an affirmative answer to the question “Are all metaphors literally paraphrasable?”166 It
should be noted that this Principle refers to the infinite potential of language to represent
rather than to the current lexicon of any given natural language; that is, people often use
metaphor precisely when they experience a gap in the lexicon, but this does not preclude the
possibility of enriching the language to include a literal expression that fills this gap.167 In
an extended section criticizing comparison views of metaphor, Searle makes the important
observation that “Sally is a dragon” seems to be a perfectly acceptable metaphor despite the
fact that dragons are fictional entities.168 In “The Logical Status of Fictional Discourse,”
Searle notes that metaphor can be employed in both fictional and non-fictional texts, and
“that what happens in fictional speech is quite different from and independent of figures
of speech.”169 It is interesting to compare Searle’s view with Martinich’s claim that fiction
involves a suspension of the supermaxim of Quality.170 These remarks are relevant to the
164
The inclusion of similarity on this list is what Searle means when he says “[s]imilarity. . . has to do with
the production and understanding of metaphor, not with its meaning” (“Metaphor” 88). Also note that by
including the part-whole relation as one of his principles of metaphor, Searle explicitly classifies metonymy
and synecdoche as species of metaphor.
165
Martinich, “A Theory for Metaphor” 456.
166
John Searle, Speech Acts (Cambridge: Cambridge UP, 1969), 88.
167
Searle, “Metaphor” 114.
168
“Metaphor” 87.
169
Searle, “The Logical Status of Fictional Discourse,” Expression and Meaning, (Cambridge: Cambridge
UP, 1979), 60.
170
Martinich, “A Theory of Fiction” 100. Martinich introduces suspension as a fourth kind of maxim
contravention. It is similar to opting out, but tends to be institutional and prolonged rather than ad hoc and
transient. Also, suspension does not preclude the performance of speech acts governed by the maxim under
48
discussion of mathematical fictionalism and its possible connexions to metaphor in chapter 4.
Though pragmatic theories of metaphor were popular in the years following Black’s original
publication, there were other noteworthy alternatives available.
Donald Davidson’s article “What Metaphors Mean” warrants attention in a survey of
metaphor theories if for no other reason than the stark contrast it provides to pragmatic
positions like those presented above. The atypicality of Davidson’s view originates in his
observation of a pervasive “central mistake” in other theories of metaphor: “the idea that
a metaphor has, in addition to its literal sense or meaning, another sense or meaning.”171
Given this observation, it is fitting that the main thesis of Davidson’s paper is “metaphors
mean what the words, in their most literal interpretation, mean, and nothing more.”172
This is not to say that metaphors are merely ornamental or otherwise problematic; rather,
Davidson explicitly explains to his readers that “[m]etaphor is a legitimate device not only in
literature but in science, philosophy, and the law; it is effective in praise and abuse, praise and
promotion, description and prescription.”173 How is it possible that metaphorical utterances
are legitimate and useful without having a meaning over and above the typically false literal
interpretation? Davidson’s explanation depends on “the distinction between what words
mean and what they are used to do”; insofar as his view assigns metaphor “exclusively to
the domain of use,” it might be classified as a pragmatic theory, though this would radically
distort the traditional understanding of that taxon.174 Additionally, “theory” may be a
bit strong: Davidson admits elsewhere that “What Metaphors Mean” does not provide a
general theory that distinguishes metaphor from other tropes and explains how metaphorical
interpretations are prompted (though he does suggest that the typical patent falsity of the
sentence in question may play a role).175 Most of Davidson’s article is dedicated to criticizing
positions that have made the “central mistake.” The little bit of positive theory Davidson
does provide culminates in the claim that “[m]etaphor makes us see one thing as another
consideration; thus, like Searle, Martinich allows for the use of metaphor within fictional works.
171
“What Metaphors Mean” 435.
172
Davidson, “What Metaphors Mean” 435.
173
Davidson, “What Metaphors Mean” 436.
174
Davidson, “What Metaphors Mean” 436. Note that Davidson does not use the word “pragmatic” to
refer to his own view. Camp and Reimer refer to Davidson’s view as “noncognitivist” (858–9).
175
Donald Davidson, “Reply to Oliver Scholz,” Reflecting Davidson: Donald Davidson Responding to an
International Forum of Philosophers, Ed. Rolf Stoecker (New York: de Gruyter, 1993), 172.
49
by making some literal statement that inspires or prompts the insight.”176 This insight,
a kind of seeing-as, is typically not propositional in nature, is potentially limitless, and
requires creativity on par with that necessary for metaphor construction.177 The scantness of
Davidson’s positive theory seems to be an asset insofar as it better allows for the possibility
of his perspective being compatible with theories of metaphor that manage to avoid his
criticisms.
There are a few features of Davidson’s criticism that are worth noting. First, he says that
metaphor cannot be a matter of ambiguity, insofar as “we are seldom in doubt that what we
have is a metaphor.”178 When genuine ambiguity or simultaneity exists regarding multiple
meanings for a sentence, the resultant figure is a pun, not a metaphor.179 These ambiguities
between multiple meanings for a word or sentence are sometimes based in dead metaphor,
when previously popular metaphorical interpretations become literally concretized. Second,
a substantial portion of “What Metaphors Mean” is dedicated to a useful comparison of
similes and metaphors: “We can learn much about what metaphors mean by comparing them
with similes, for a simile tells us, in part, what a metaphor merely nudges us into noting.”180
Davidson considers two ideas regarding the connexion between simile and metaphor. The first
is the traditional “metaphors are elliptical similes” view that dates back to Cicero, the second
176
Davidson, “What Metaphors Mean” 445.
Davidson, “What Metaphors Mean” 444, 435. John Austin, the father of speech act theory, provides
a terminological framework that may be helpful here. Austin argues that an instance of saying something
can be seen as comprised of three interconnected but distinguishable acts. The locutionary act has phonetic
(sound making), phatic (construction according to language rules), and rhetic (sense and reference imbued)
components, and is the locus of meaning (Austin 95). The illocutionary act is the purposive component of
the utterance, the intended conventional force of the utterance (Austin 109). The perlocutionary act is what
the utterance actually accomplishes, whether intended or not (for example, uttering a threat may variously
subdue, provoke a fight, incite laughter, cause disappointment, and so on) (Austin 118). In differentiating
speaker and sentence meaning, Searle seems to emphasize the locutionary component of the metaphorical
utterance; the point of the speech act is the transmission of the metaphorical speaker meaning to the listener.
Davidson, on the other hand, seems to emphasize the perlocutionary aspect of the metaphorical utterance
(being prompted to a seeing-as) while maintaining that the locutionary act is unchanged whether a sentence
is uttered metaphorically or literally. Bearing in mind that Austin explicitly acknowledges that he does not
account for non-literal language use, Davidson’s theory of metaphor seems consistent with Austin insofar as
it does not overcomplicate the locutionary act and emphasizes the non-conventional nature of metaphorical
perlocution (Austin 119, 122).
178
Davidson, “What Metaphors Mean” 437.
179
Davidson, “What Metaphors Mean” 437. Similarly, Davidson later distinguishes lies from metaphors
(“What Metaphors Mean” 442). Some elaboration on these points is desirable insofar as metaphoric language
seems to be able to deceive and to be used in the service of some puns. This observation does not crucial
enough to warrant specific examples.
180
Davidson 439.
177
50
is a sophisticated variation on this idea: the metaphorical meaning of a sentence is identical
with the literal meaning of a corresponding simile, for some notion of “corresponding.”181
He rejects both of these views as incorrect based on the observation that similes are simple
to understand insofar as “everything is like everything, and in endless ways,” whereas many
metaphors are “very difficult to interpret and, so it is said, impossible to paraphrase.”182 It
may be noteworthy that Davidson does not seem to immediately reject Nelson Goodman’s
suggestion that “the difference between metaphor and simile is negligible.”183
Arguably the biggest problem with “What Metaphor Means” is its failure to provide any
discussion of what is meant by “literal meaning”; if the article is considered as a standalone
piece, this is a grievous oversight. In particular, it seems to be the key factor behind two of
Oliver Scholz’s criticisms of Davidson’s theory: he claims that it is unable to accommodate
both metaphors relying on the false stereotypes that make up the “system of associated commonplaces” and metaphors containing fictional terms.184 Related to this point is a weakness
Davidson acknowledges: his theory is not equipped to deal with non-assertive speech acts
such as questions and commands.185 However, a significant proportion of Davidson’s other
writings are dedicated to considerations of meaning that can potentially rectify these shortcomings. In “Truth and Meaning,” Davidson argues that the Tarskian definition of truth
satisfies the sufficiency conditions for a theory of meaning. Davidson’s theory of meaning is
a radical departure from earlier theories of meaning in that it consists only of Tarskian Tsentence theorems relating sentences in the object language to sentences in the metalanguage
and therefore has no use for meanings per se, understood as ontologically distinct entities.186
In particular, it is worth noting the drastic disparity between this conception of meaning and
Grice’s notion of nonnatural meaning. Davidson briefly foreshadows his theory of metaphor
in “Truth and Meaning”: “When we depart from idioms we can accommodate in a truth
181
Davidson, “What Metaphors Mean” 439.
Davidson, “What Metaphors Mean” 440; emphasis mine.
183
Davidson, “What Metaphors Mean” 440.
184
Oliver Scholz, “ ‘What Metaphors Mean’ and how Metaphors Refer,” Reflecting Davidson: Donald Davidson Responding to an International Forum of Philosophers, Ed. Ralf Stoecker (New York: de Gruyter, 1993),
165.
185
Davidson, “What Metaphors Mean” 442.
186
Donald Davidson, “Truth and Meaning,” Synthese 17 (1967): 101–2. It is worth noting that Max Black’s
discussion of meaning in chapter two of Models and Metaphors is quasi-Tarskian and also concludes that
there are no entities called meanings (24).
182
51
definition, we lapse into (or create) language for which we have no coherent semantical account.”187 It thus seems reasonable to assume that when Davidson says there is no such
thing as metaphorical meaning, he is claiming that metaphorical interpretation cannot be
reduced to necessary and sufficient truth conditions. In later writings, Davidson apparently
recants some of his claims about meaning: “In my essay ‘What Metaphors Mean’. . . I was
foolishly stubborn about the word meaning when all I cared about was the primacy of first
meaning.”188 By “first meaning,” Davidson means something roughly correspondent to literal
meaning, the kind of meaning that can be found in dictionary entries; he implemented this
jargon in hopes of distancing himself from various negative associations that “literal” has. In
at least two articles (“A Nice Derangement of Epitaphs” and “Locating Literary Language”)
Davidson uses the notion of first language to roughly sketch an account of implicated “meanings” that is similar to — but also markedly different from — that of Grice.189 In particular,
Davidson comments that Grice’s principles of implicature do not seem sufficient to handle
all of the cases he wishes to address (notably, malapropisms and literature), and that they
curiously lack a speaker intention factor. Interestingly, the nature and potential proliferation
of malapropisms leads Davidson to ultimately conclude that “we should give up the attempt
to illuminate how we communicate by appeal to conventions.”190 This conclusion seems to
suggest the existence of a rift between communication and formalization, an idea that is
certainly worthy of further consideration. Davidson’s theory of meaning is intriguing and
relevant, but also expansive and complex; exploring a portion of the contextual corona of
“What Metaphors Mean” has been usefully elucidatory, but delving further into his account
of meaning would be too tangential to the thrust of this chapter.
This chapter has presented a philosophical history of the concept of metaphor, from its
ancient Greek origins where it referred to physical translocation and its Aristotelian meaning
of a transfer of words, through centuries of disregard, slander, and hypocritical appeals for the
187
Davidson 106. This passage may be usefully relevant to considering the relationship between idiom and
metaphor.
188
Donald Davidson, “Locating Literary Language,” Truth, Language, and History (New York: Oxford UP,
2005), 173.
189
Though he has acknowledged that one might legitimately use the word “meaning” for both first meaning
and speaker meaning, Davidson continues to avoid the latter usage in his own writings.
190
Donald Davidson, “A Nice Derangement of Epitaphs,” Truth, Language, and History (New York: Oxford
UP, 2005), 107.
52
total elimination of metaphor, and culminating in the twentieth-century renaissance in the
philosophy of language and its myriad attempts to provide a positive, legitimating, rigorous
account of metaphor by shifting the focus of inquiry from words to word-meanings, sentencemeanings, speaker-meanings, or beyond. The purpose of this undertaking was to provide
a motivational and foundational context for the discussion of conceptual metaphor theory
in chapter 3 and, in turn, support the discussion of the connexions between mathematics
and metaphor in the remaining chapters. While there are hundreds more published accounts
of metaphor that could be considered, the few discussed here constitute a sufficient sample
insofar as they are among the most influential viewpoints and are jointly representative of
the diversity in this field. None of the above theories seems obviously superior to the others,
and the lack of consensus in the academic community is a testament to this. However,
there are many insights into metaphor scattered throughout this chapter that range from the
ubiquitous and nearly inane to the novel and nearly profound.
First, an obvious observation: metaphors do exist. That is, one encounters utterances
or expressions or sentences in both spoken language and written text that are identified as
metaphorical. Hence, any complete account of metaphor must be somehow connected to
an understanding of language. However, the identification of metaphor is not as straightforward, obvious, and non-controversial a process as many theorists would have us believe.
The application of the term “metaphor” seems somewhat obvious in a general sense, but
becomes difficult when one begins to consider the details rigorously. Some metaphors are
easily detected and identified, while others may only be recognized in retrospect. Once a
metaphor is identified, the identifier may or may not be able to explain their reasoning
for arriving at that conclusion. One widespread strategy, though it constitutes neither a
necessary nor a sufficient condition, is the identification of falsity, absurdity, or some other
violation of the conventions of language. While recognition of metaphor plays a crucial role
in most traditional accounts of metaphor interpretation, some — notably Davidson’s — do
not seem to require recognition for efficacy. Intimately connected to the issue of identification
is the observation that linguistic meanings are context dependent: no sentence is definitively
metaphorical (or non-metaphorical) apart from contextual considerations. These issues surrounding the identification of linguistic metaphors are one of the motivations for considering
53
conceptual theories in the next chapter: the intuition that mathematics involves metaphor
conflicts with the fact that mathematical sentences such as “Odd numbers are not divisible
by two” are rarely, if ever, identified as metaphorical.
There are several other points arising from the above discussion that a theory of metaphor
should seek to account for. Every view of metaphor presents it as some kind of relation, between vehicle and tenor or figure and frame, for example. A fundamental issue is whether this
relation is understood to be symmetrical or not. The current trend holds that metaphor is, in
general, asymmetrical: “That surgeon is a butcher” and “That butcher is a surgeon” seem to
express two very different thoughts, and “The sun is Juliet” seems nearly incomprehensible,
for example. One’s answer to the symmetry question will have widespread ramifications for
their theory, including implications for how metaphor may relate to other figures and tropes,
such as simile. Another important point is that the idea of “dead metaphor” is ubiquitous
in metaphor theories. This suggests a pervasive belief in a connexion between metaphor and
linguistic development and evolution exists, a belief which merits attention. A final related
point that theories of metaphor should be cognizant of is that metaphor itself is frequently
understood by way of metaphors — what might be called metametaphors. Metaphors were
originally conceived metaphorically as a “carrying across,” and contemporary authors continue to invoke metaphor in discussing metaphor (such as when they speak of dead metaphor,
for example). Metametaphors may be taken to be abhorrent, acceptable as a useful communicative device, or ineliminably constitutive of our understanding of metaphor, but they
should not be ignored. Symmetry of metaphor, metaphor death, and metametaphoricity are
discussed as central issues in conceptual metaphor theory.
Despite the disparities between the views presented in this chapter and the disagreements
they lead to, there is a unifying thread running through most of them. Whether semantic or
pragmatic, descriptivist or referentialist, almost every view above subscribes to at least some
of the core beliefs of the literal-truth paradigm. That is, each subsequent development in the
metaphor theory in this chapter has been analogous to the addition of a corrective epicycle
to a fundamentally flawed theory of celestial motion. As such, a certain set of objections
applies pervasively:
[Most theories considered thus far] take falsity or anomaly as a criterion that
54
allows for the recognition of metaphor, and they all think of metaphorical interpretation as secondary, and based on processes that are rather different from
those governing the interpretation of literal language. Moreover, all approaches
[considered thus far] have problems with novel metaphors and the apparent ‘creation of similarity.’ In large part, these difficulties stem from the assumptions
that literal language has an absolute priority over figurative language and that
metaphor is essentially deviant, or at least distinct, from the literal.”191
Additionally, most theories of metaphor in the literal-truth paradigm fail to discuss the possibility of non-linguistic metaphors, including those occurring in other media (paintings, for
example), and more radical alternatives such as Nicomachus’ cross-disciplinary application
of skills (discussed near the beginning of this chapter).192 Such objections are largely responsible for the advent of conceptual metaphor and the paradigm shift it causes. Theories
of conceptual metaphor grow out of the seeds planted by earlier skeptics of the literal-truth
paradigm — including Vico, Nietzsche, and especially Richards — and are the subject of
chapter 3. The development of the notion of conceptual metaphor is arguably the most important advent in metaphor theory from the perspective of someone interested in metaphor
and mathematics. This is because the traditional view of mathematics as the ideal form of
language use is as connected to the literal-truth paradigm as the traditional hostile views
regarding metaphor are. In providing a serious alternative to the literal-truth paradigm,
theorists simultaneously managed to empower and legitimate metaphor while making mathematics less untouchable, and thereby moved the two subjects close enough together for them
to constructively associate for the first time.
191
Leezenberg 135.
It should be noted that the absence of such explanations does not necessarily entail that none of the
positions above can account for these phenomena.
192
55
Chapter 3
Conceptual Metaphor
The locus of metaphor is not in language at all, but in the way
we conceptualize one mental domain in terms of another.1
— George Lakoff
Chapter 2 explored some of the ways in which traditional theories of metaphor are problematic. The thesis common to most of these accounts is that metaphors are essentially
linguistic phenomena and are necessarily derivative from established literal language. As I
showed in chapter 2, this premise can lead to the problematic view that metaphors necessarily
deviate from the truth and are unnecessarily obfuscating or merely ornamental, a sentiment
that arguably reached its zenith with the logical positivist movement.2 Perhaps more significantly, the fundamental place of “literality” in such theories means their proponents owe an
account of this notoriously difficult notion. While a handful of scholars doggedly attempt to
provide solutions to these problems from within their linguistic perspectives on metaphor, a
new generation of metaphor researchers have instead adopted a radically different approach
to metaphor.
A paradigm shift has been occurring in metaphor scholarship. A substantial number of
contemporary metaphor scholars consider metaphor to be fundamentally a matter of thought
rather than a primarily linguistic phenomenon. Although this approach has only gained popular support within the last few decades, it did not emerge ex nihilo. As suggested in chapter
2, notable progenitors of this hypothesis include Aristotle, Vico, Nietzsche, I.A. Richards,
and Max Black. In particular, Richards might plausibly be identified as the father of this new
movement based on his pronouncement that “[t]hought is metaphoric. . . and the metaphors of
1
George Lakoff, “The Contemporary Theory of Metaphor,” Metaphor and Thought, 2nd ed., Ed. Andrew
Ortony (New York: Cambridge UP, 1993), 203.
2
Skorupski 65–9.
56
language derive therefrom.”3 Many factors certainly contributed to the development of this
new trend, but three are notably important to its emergence in the late 1970s as a serious
alternative to the traditional view. First, none of the various linguistic theories of metaphors
managed to prove themselves clearly superior to their competitors. Serious objections to each
of the main linguistic accounts were then known and, indeed, persist to this day. Such a situation can provide motivation to investigate new or overlooked avenues. Second, the middle of
the twentieth century saw the development of significantly improved empirical techniques and
methodologies that allowed cognitive scientists to perform new research into how we think
and reason. Many of these experiments generated results that were seen as incompatible with
traditional understandings of language and cognition, and thereby provided impetus for the
development of new theoretical frameworks. Third, though almost certainly related to the
second point, there was a shift in attention within the philosophical literature on metaphor
from the one-off poetically figurative sentences considered prototypically metaphoric by past
scholars to the kind of ubiquitous, everyday metaphoricity that usually goes unnoticed; I
discuss this distinction at greater length below. Though there are now multiple accounts of
metaphor which hold it to be a basic part of cognition and not merely a fortuitous kind of
linguistic failure, my attention in this chapter will be primarily focused upon the conceptual
metaphor theory (CMT) proposed by George Lakoff and Mark Johnson.
I have opted to explore this particular cognitive theory of metaphor in detail for a number
of reasons. Lakoff and Johnson were among the pioneers that ushered in this new paradigm in
metaphor scholarship.4 Their seminal book Metaphors We Live By is an ambitious attempt
to develop a novel theory of metaphor that overcomes the major problems associated with
traditional linguistic views. Written in an accessible style, this work served as many people’s
first significant exposure to the metaphor-as-thought movement. Metaphors We Live By is
3
Richards 51; emphasis his.
A few other scholars deserve mention here (besides the Newtonian giants mentioned in the previous paragraph). Lakoff and Johnson were significantly inspired by Michael Reddy’s work on the conduit metaphor
in their development of the notion of conceptual metaphor (Metaphors We Live By 10). Harald Weinrich’s
1976 text Sprache in Texten posits that “verbal metaphors are not isolated occurrences but fall into semantically homogenic groups” (Müller 48–9) and that certain abstract notions simply cannot be grasped without
using metaphors (Müller 50). Weinrich’s theory has strong parallels to Lakoff and Johnson’s work, but was
developed independent of CMT and predates it by a few years; however, it has had minimal influence in
anglophone scholarship as it has never been translated into English.
4
57
a provocative read that tends to elicit a strong reaction, and has accordingly been an important source of inspiration to many metaphor scholars, prompting support and criticism alike,
and motivating the development of a variety of alternative theories. In the thirty years since
this publication was released, Lakoff, Johnson, and a host of collaborators and advocates
have generated a significant body of work on conceptual metaphor, its relationship to the
embodied mind, and the implications of both. Conceptual metaphor theory is thus worthy
of consideration as one of the most-established, best-known, and substantial — even if thoroughly controversial — theories of metaphor within the new cognitive paradigm. However,
from the point of view of this dissertation, the principal reason for investigating CMT is that
it forms the basis of the most important attempt to relate metaphor and mathematics yet
published. Where Mathematics Comes From, co-written by Lakoff and Rafael Núñez, is a
tour de force account of how the various branches of advanced abstract mathematics develop
out of primitive cognitive capacities by means of conceptual metaphor. This chapter will be
devoted to an examination of Lakovian CMT in general, while one of the foci of chapter 4 is
to assess the account of mathematics grounded in this theory of metaphor.
It has been said that “[w]ithout concepts, there would be no thoughts.”5 If this plausible
claim is accepted, then it follows that any theory of metaphor-as-thought must involve concepts, whether by relying on some specific account of concepts, having implications for how
concepts are understood, or through some combination thereof. As is evident from its name,
CMT is no exception: concepts are fundamental to CMT given its claim that our mental lives
are extensively shaped by conceptual metaphors, cognitive mappings that allow structure to
be constitutively imported into one conceptual domain from another.6 This posited deep
connexion between metaphors and concepts is also the locus of many of the objections to
the Lakovian approach. These tend to fall into two categories. The first type argues that
the understanding of concepts entailed by the Lakovian notion of conceptual metaphor is
unsatisfactory. The second type involves concerns about the relationship between concepts
5
Jesse Prinz, Furnishing the Mind: Concepts and their Perceptual Basis (Cambridge: MIT Press, 2002),
1.
6
Note that Lakoff et al. “use the term cognitive in the richest possible sense, to describe any mental
operations and structures that are involved in language, meaning, perception, conceptual systems, and reason”
(Lakoff and Johnson, Philosophy in the Flesh 12). I mention this to avoid confusion, as many philosophers
adopt a narrower understanding of “cognitive.”
58
and language espoused by CMT. Given the centrality of concepts to CMT, I start by briefly
introducing the notion of “concept,” thereby providing necessary context for understanding
CMT. I follow this with a synopsis of CMT, starting with its roots in Metaphors We Live By
and tracing key developments to the theory that have emerged over the last three decades.
The final section of the chapter presents a defense of CMT against a variety of objections.
My objective is to defend the general claim that metaphor is not a merely linguistic phenomenon, not to resolve the technical differences between apparently similar accounts of
metaphor-as-thought. Inspired by the work of Cornelia Müller, I conclude by suggesting that
in general, metaphor can not be reduced to either the conceptual or to the linguistic but is
best understood as involving the dynamic interplay between them.
3.1
Theories of Concepts
“Concept” is a complicated and controversial term with a long and varied philosophical
history; it is beyond the scope of my project to provide a comprehensive analysis of it.7 There
are many competing philosophical accounts of concepts in the literature, none of which is
decisively victorious. Rather than surveying the many accounts of concepts, I adopt a general
perspective in order to bypass controversial details and focus on aspects of concepts there is
some agreement about. This general discussion provides a common ground for considering
Lakoff’s specific theory of concepts. Concepts are that which is essential to most human
cognitive activity; for example, the concept BIRD is utilized when a person speaks about some
bird, recognizes something as a bird, reasons about birds, etc. The word “concept” is used
in everyday English to refer both to specific understandings of things held by individuals,
as well as to collective understandings associated with groups of individuals, often without
recognizing a distinction between these two.8 However, a more definite understanding of
7
Readers whose interest is not sated by my brief discussion can find many works written by philosophers
and cognitive scientists on the topic of concepts. A few noteworthy examples include Jerry Fodor’s Concepts:
Where Cognitive Science Went Wrong (1998), Gregory Murphy’s The Big Book of Concepts (2002), Jesse
Prinz’s Furnishing the Mind: Concepts and Their Perceptual Basis (2002), and the Eric Margolis and Stephen
Laurence-edited volume Concepts: Core Readings (1999).
8
In this chapter, I avoid using the word “concept” in a non-technical sense as much as possible to avoid
confusion. I will make it clear, either explicitly or by way of context, if I am using the term to refer to some
specific understanding of “concept.”
59
concepts than this is desirable in philosophy, where the notion plays important explanatory
roles in theories of mind, language, and meaning. Philosophers have been theorizing about
concepts almost as long as they have been using them, trying to clear up some of the ambiguity
and imprecision found in the common understanding. Indeed, that the attempt to bring
heightened clarity and rigor to a notion is referred to as conceptual analysis indicates how
central the idea of “concept” is to much of philosophy.9 How conceptual analysis proceeds
will depend upon which theory of concepts the philosopher holds.
Navigating the bewildering assortment of conceptual theories can be a daunting task.
Thankfully, help is available. Jesse Prinz’s seven desiderata for a theory of concepts provide
a useful framework for thinking about the relative merits of competing accounts; I will use
these criteria to evaluate the classical theory of concepts and compare it to some leading
alternatives.10 Two of Prinz’s desiderata are primarily concerned with requiring theories to
accommodate empirical data. The first desideratum concerns the scope of a theory. Concepts
are observed to come in a wide variety of flavours: they possess varying levels of abstraction;
they may be theoretical, or formal, or natural; they may be fleeting or persistent; etc. Any
theory of concepts must either be able to account for this observed variation in the kinds of
concepts we are able to possess or to somehow explain it away.11 The second desideratum
acknowledges that a connexion exists between concepts and how we categorize the world and
therefore an adequate theory of concepts ought to fit current empirical data on categorization.
Cognitive scientists — notably Eleanor Rosch — have studied both category identification
(identifying the category that an object belongs to) and category production (recognizing
what attributes an object possesses given that it belongs to a certain category) producing
data relevant to theories of concepts; I will discuss the specific observed effects below.12
Three of Prinz’s desiderata deal explicitly with the content of concepts. The intentional
9
I am using “conceptual analysis” very broadly here to refer to the practice of attempting to improve our
understanding of some concept through philosophical discussion and argumentation rather than the narrower,
more technical usage found in early analytic philosophy.
10
Other authors give conditions they believe a theory of concepts must satisfy (for example, chapter 2 in
Fodor (1998) contains “five non-negotiable conditions on a theory of concepts” (23)). I have chosen to use
Prinz’s because they seem suitably broad in scope (they subsume Fodor’s conditions, for example) and are
relatively impartial in tone.
11
Prinz 3.
12
Prinz 9–10
60
content desideratum states that a theory of concepts should account for how concepts come
to “represent, stand in for, or refer to things other than themselves.”13 To meet this condition, a theory will probably have to say something about what these intentional contents are,
whether they are understood to be natural kinds, ad hoc categories, or some other thing.14
Intentional content does not seem to exhaust conceptual content, as shown by various puzzles of reference, notably those that inspired Frege’s distinction between Sinn (sense) and
Bedeutung (nominatum) as well as Hilary Putnam’s Twin Earth scenario. Prinz therefore
includes a desideratum of cognitive content that states a theory should explain how concepts
with identical intentional content can differ and how concepts with different intentional content can be alike.15 The final desideratum of content claims that the apparent unbounded
productivity of our cognitive capacities has its root in the compositionality of our concepts.
Compositionality is the idea that we possess many compound concepts and that the content and structure of such concepts functionally depends upon their constituent concepts.
A finite number of combination rules for forming compound concepts together with a finite
number of primitive concepts would theoretically allow for the creation of an infinite number
of concepts.16 Prinz clarifies that this criterion requires that both intentional and cognitive
content be compositional.17 The compositionality condition is central to the dispute between
contemporary theories of concepts.
Prinz’s two final desiderata have to do with the active life of concepts. The acquisition
desideratum posits that a theory of concepts should provide — or at least be compatible with
— some credible account of how humans acquire concepts. This explanation must account
for both the ontogeny and phylogeny of concepts if the two are distinguishable in the given
theory. At the core of the acquisition condition lies the question of whether concepts are
innate or learned.18 Intimately related to questions of concept acquisition is the thought
that concepts must be public. The publicity desideratum says that both the intentional and
cognitive content of a concept “must be capable of being shared by different individuals and
13
Prinz
Prinz
15
Prinz
16
Prinz
17
Prinz
18
Prinz
14
3.
4.
7–8.
12–3.
14.
8–9.
61
by one individual at different times.”19 This is a necessary condition if concepts are to play
an explanatory role in linguistic communication and in intentional behaviour.20
Prinz considers two further desiderata regarding the relation of concepts to language,
but ultimately rejects them. The first is that “concepts simply are the meanings of words,”
and Prinz abandons this because it rules out popular reference-based semantic theories.21
The second is that “public language is necessary for the possession of concepts,” a claim
based in Wittgenstein’s private language argument that is controversial among most cognitive
scientists and many philosophers.22 As Prinz wants to keep open the possibility that “one can
present a theory of what concepts are without mentioning language,” he does not include
any desiderata specifically about language in his list.23 However, if CMT is going to use
conceptual metaphors to account for the metaphors we observe in language then it must
necessarily say something about the connexion between language and concepts. Keeping this
proviso in mind, Prinz’s seven desiderata provide a framework that can be used to evaluate
the classical theory of concepts, as well as subsequent views, including CMT.24
From the time of Socrates until the last century, most scholars based their analyses of
concepts in some version of what is now called the classical theory of concepts. The classical
theory of concepts is, in fact, a “diverse family of theories centered around the idea that
concepts have definitional structure.”25 The idea that concepts are essentially equivalent
to definitions — exhaustive lists of necessary and sufficient membership criteria — remains
influential to this day. Because membership criteria for any given concept are given in terms
of other concepts, a systematic hierarchy of concepts exists in the classical theory. One
19
Prinz 14.
Prinz 14–5.
21
Prinz 17.
22
Prinz 18.
23
Prinz 21.
24
Prinz’s desiderata are intended primarily as a framework for assessing whether a theory adequately
accounts for conceptual structure, and provide only minor constraints on ontological concerns. I too am
mostly interested in conceptual structure and am more or less indifferent to ontology, that is to whether
concepts are mental representations or cognitive capacities, for example. That being said, I am leery of
ontologies positing conceptual atomism (insofar as it is incompatible with conceptual metaphor) or concepts
as abstract entities (as counterproductive to attempts to explain away abstract mathematical entities). For
the purposes of this dissertation, I must therefore reject any theory of concepts that assumes or implies either
of these positions.
25
Stephen Laurence and Eric Margolis, “Concepts and Cognitive Science,” Concepts: Core Readings, Eds.
Eric Margolis and Stephen Laurence (Cambridge: MIT Press, 1999), 9–10.
20
62
can already see traces of the classical view in Plato’s writings: in several dialogues, Socrates
engages various characters in a collaborative conceptual analysis aimed at limning the essence
of JUSTICE, PIETY, LOVE, etc. It is clear from the definitional nature of concepts in the classical
view that they are intimately connected with categories; for example, defining a triangle as
a polygon with exactly three sides gives membership criteria for the class of triangles. In
Plato’s version of the classical view, if human beings ever possess true knowledge then a deep
connexion must exist between at least some mental concepts and the eternal Platonic Forms,
as instantiated triangles in the world are related by virtue of being mere approximations
of the same Form. The ancient Greeks and Romans combined Plato’s philosophy with the
theory of categorical logic developed in Aristotle’s Organon and established the classical
view of concepts that was dominant for two millennia. Later philosophers held variants of
the classical view that differed from that espoused by the ancients but retained the central
role of definition; for example, Locke’s theory of ideas differs starkly from Plato’s theory of
Forms but the differences shroud a common core. It is a testament to the staying power of
the classical theory that the contemporary English word “concept” has undergone relatively
little change in meaning since its origin in the Latin term conceptum.26
Classical theories do an excellent job of satisfying many of Prinz’s criteria, which perhaps
helps explain their long reign. Laurence and Margolis suggest that the classical theory offers
“unified accounts of concept acquisition, categorization, epistemic justification, analytic entailment, and reference determination, all which flow directly from its basic commitments.”27
Prinz observes that the classical theory also appears to satisfy the cognitive content and
publicity conditions.28 The idea that concepts are definitions is a powerful one that does a
lot of explanatory work both elegantly and economically.29 Unfortunately, there are many
26
“concept, n.” OED Online, Mar. 2014, Oxford UP, 25 Apr. 2014. The noun conceptum derives from
the verb concipere, a word whose primary meaning was “to take inside,” as in the conceiving of a child,
but that also sometimes meant “to perceive by means of the senses,” “to express in formal language, frame,
draw up,” and, most relevant to current considerations, “to conceive or grasp in the mind, form an idea of,
imagine.” All of these meanings were known in antiquity; the latter usage was available at least as early
as the first century BCE, as it can be found in Cicero’s writing (“concipiō”, Oxford Latin Dictionary). The
metaphor extending the idea of pregnancy from the physical to the mental sphere, allowing concepts and
ideas to be understood as offspring of the mind, can be traced back at least as far as Plato’s writings; it is
seen in Socrates’ tale of Diotima’s teachings in the Symposium, for example (Plato, Symposium).
27
Laurence and Margolis 10.
28
Prinz 38.
29
For example, in the classical theory, the concept LITERAL is composed of necessary and sufficient conditions
63
reasons to think that the classical theory of concepts is untenable. One strong reason to
reject the classical theory of concepts goes all the way back to Plato: it seems difficult, if
not impossible, to come up with adequate definitions for most concepts; as Laurence and
Margolis put it “there are few, if any, examples of definitions that are uncontroversial.”30
While definitions for concepts like LOVE and JUSTICE are perennially controversial, even seemingly straightforward definitions such as “a bachelor is an unmarried man” can be seen as
problematic under mild scrutiny (is the Pope a bachelor? A widower? A man in a committed monogamous common-law relationship? A single 15-year-old male? A single 25-year-old
transsexual?). Wittgenstein’s discussion of family resemblance concepts in his Philosophical
Investigations also provides relevant evidence against the classical view. Famously, Wittgenstein argued that the concept
GAME
does not admit of a necessary condition, that there is no
essence that all games have in common. Rather, he believed that
GAME
and similar concepts
are unified as a chain is, with each constituent part overlapping some others without overlapping all others.31 Such considerations show that the classical theory does not satisfy the
scope requirement because “most of the concepts we use are impossible to define.”32 Another
reason to be apprehensive about the classical theory comes from Quine’s “Two Dogmas of
Empiricism.” The view that concepts are definitions seems incompatible with Quine’s conclusions about analyticity: “If there is no principled way to distinguish analytic beliefs from
collateral knowledge, definitions devolve into unwieldy, holistic bundles. . . [t]hus, until definitionists provide a principled analytic/synthetic distinction, they have difficulty explaining
how concepts can be shared.”33 These philosophical objections revolve around the idea that
many concepts possess an indeterminacy that seems fundamentally incompatible with their
being definitions. Based on these objections and others, many philosophers have abandoned
the classical theory of concepts.34
and, therefore, my worries about the vagueness of the notion would seem to be unfounded!
30
Laurence and Margolis 15.
31
Ludwig Wittgenstein, Philosophical Investigations, Trans. G.E.M Anscombe (Oxford: Basil Blackwell,
1972), 31e –32e .
32
Prinz 41.
33
Prinz 41. Scholars differ on the strength of this objection depending on their interpretation of Quine’s
argument (Laurence and Margolis 20–1).
34
It should be noted that a rejection of the classical view of concepts typically does not necessarily entail
abandonment of definitions in general. Coming up with good definitions for terms can be exceedingly fruitful
even if we acknowledge that those definitions will rarely exhaust the concepts associated with those terms.
64
The community of cognitive scientists has been nearly unanimous in its rejection of the
classical theory, due in large part to the experimental findings of Eleanor Rosch. If the
classical theory is true, then the necessary and sufficient membership conditions comprising a
concept should form a clear, well-defined partition between members and non-members of the
associated category. However successful definitions may be at distinguishing members from
non-members of a category, they do not seem equipped to make distinctions between any two
members of their category; that is, a definition can only define the boundary of a category, it
cannot provide any internal structure. In the early 1970s, Rosch and her colleagues performed
a series of groundbreaking experiments demonstrating that many concepts possess an internal
structure in the form of a typicality gradient, a fact that most psychologists have taken to be
fatal to the classical theory. This research asked subjects to categorize a variety of exemplars,
rating the typicality of each category member. A comparison of the reported typicality with
the measured speed of answer showed that people do not take all members of a category to
be on a par with each other, but rather take some members as more typical representatives
of a class than others. For example, given the category
BIRD,
people tend to judge robins
as very typical representatives, chickens and vultures as moderately typical, and ostriches
and penguins as atypical.35 In addition to showing that intracategorial structure is graded
rather than homogeneous, Rosch found that intercategorial structure exists: just as some
members of a given category are more typical than others, some categories are more basic
than others. Experiments show that, given an object, people are likely to categorize it at an
intermediate level between the general and the specific. For example, people are typically
faster to recognize an object as a dog than as a spaniel or an animal. Research also shows that
these basic-level concepts seem to be acquired earlier in development.36 Because the classical
theory does not seem to be able to explain typicality effects or basic-level categorization, it
fails to satisfy the categorization desideratum and, arguably, the acquisition criterion as well.
Rosch’s findings about categorization were thus an important source of motivation for Lakoff
in his development of CMT.37
The classical theory of concepts fails because it does not adequately satisfy at least the
35
Laurence and Margolis 24–5.
Prinz 10.
37
George Lakoff, Women, Fire, and Dangerous Things (Chicago: U of Chicago P, 1987), 15.
36
65
scope and categorization conditions. In the wake of Rosch’s research, several alternative
theories have arisen and been put to good use, though no one of them has emerged as clearly
superior. The first serious alternative approach was prototype theory, a cluster of viewpoints
unified by the idea that “most concepts — including most lexical concepts — are complex
representations whose structure encodes a statistical analysis of the properties their members tend to have.”38 According to prototype theory, a concept is a mental representation
of an idealized instance that possesses the maximal number of typical properties for some
category.39 Intentional content is defined by resemblance to the prototypical representation;
thus, the possession of disjoint subsets of typical properties may be sufficient for category
membership.40 Prototype theory was strongly motivated by Wittgenstein’s discussion of family resemblance and Rosch’s research into typicality effects.41 It is no surprise, then, that one
of its major strengths is being able to account for the typicality effects that help comprise
the categorization desideratum. Concept members possessing a larger allotment of typical
features tend to be identified as better representatives of that concept, and thus graded intracategorial structure is explained by prototype theory. The abandonment of strict necessary
and sufficient membership conditions makes prototype theory impervious to the main criticisms of classical theory rooted in definitional structure, yet it has at least two weaknesses
of its own. First, many concepts seem to lack prototype structure and therefore prototype
theory arguably fails to satisfy the scope desideratum. The most accessible examples may
be complement concepts such as
NOT-A-DOLPHIN
(is a hamburger a more typical not-dolphin
than a saxophone, a crow, or a plastic scale model of a dolphin?), but many other examples exist including empty or uninstantiated concepts such as
ANCIENT ROMAN SPORTSCAR.
42
Second, prototype theories seem to have difficulty meeting the compositionality condition as
suggested by the following example: mercury is arguably the prototypical
LIQUID METAL,
it is a relatively atypical instance of
LIQUID
METAL
(compared to, say, iron) and of
38
yet
(compared
Laurence and Margolis 27.
Prinz 52.
40
Prinz 75.
41
Laurence and Margolis 28.
42
Laurence and Margolis 35–6. One possible line of defense is to deny that complement concepts based
upon a negation are genuine concepts based on the artificiality of their construction. However, it does not
seem likely that this defense will work for all conceivable examples.
39
66
to water).43 While some scholars attempt to amend prototype theory to overcome these
criticisms, others take them to be fatal and thus have sought other approaches.
A more recent alternative to the classical and prototype theories is the theory-theory of
concepts.44 This approach is built around an analogy between how concepts are interrelated
and how the terms in a scientific theory are related; more specifically, the theory-theory
holds that a concept “is constituted by its role in an explanatory schema.”45 Concepts are
thus “mini theories of the categories they represent.”46 Theory-theory occupies an appealing
middle ground between the rigid necessary and sufficient conditions of the classical theory
and the loose correlational similarity condition of the prototype theory. Through its core
analogy, theory-theory allows the voluminous literature on scientific theorization to inform
our understanding of concepts. Its major advantages over prototype theory involve being
better able to account for concepts with category membership based on more than mere
resemblance, such as unobservable properties and causal connectedness.47 Theory-theory
emphasizes causal and explanatory relations among category features, and is thus better
able to incorporate the observed human tendency towards essentialization in classification
than classical and prototype theories are. Theory-theory also has the advantage of being able
to import ideas about scientific theory development to account for conceptual development
in humans over time.48 However, theory-theory has some difficulties in accounting for the
intentional content of concepts (“How do theories refer to categories?”), their cognitive compositionality (“[t]heories, in all their cumbersome complexity, are not the kinds of things that
can be easily combined”), and their publicity (“[i]t is very unlikely that any two people have
exactly the same theories of the categories they represent”).49 While the theories discussed
thus far have differed in the main only on the kind of conceptual structure they posit, others,
43
Laurence and Margolis 39. The compositionality criticism of prototype theory has frequently been
championed by Fodor. Prinz, chapter 11, argues that prototypes exhibit a version of compositionality that
is weaker than Fodor requires but is sufficient to satisfy the desideratum.
44
The theory-theory of concepts should not be confused with the theory-theory of mind. Gregory Murphy, a
notable advocate of a version of the theory-theory of concepts, refers to his view as the “knowledge approach”
to avoid this potential confusion (Murphy, The Big Book of Concepts 61).
45
Laurence and Margolis 45.
46
Prinz 76.
47
Prinz 77.
48
Laurence and Margolis 45–6.
49
Prinz 86–7.
67
such as Fodor’s atomism, take a totally different approach.
Like the theory-theory, conceptual atomism emerges in reaction to the failures of the
prototype and classical theories, though the solution it proposes is more radical.50 Conceptual atomism holds that lexical concepts have no structure whatsoever; they are primitives
that do not decompose into properties or features but have a content “determined by the
concept’s standing in an appropriate causal relation to things in the world.”51 The strengths
of conceptual atomism include naturally satisfying the intentional content desideratum in a
way that bypasses concerns regarding error that plague other theories. Conceptual atomism
asserts that the causal relation between a concept and its instances allow for direct reference,
thus avoiding complications caused by reference-mediating entities such as definitions, prototypes, and theories.52 Its rejection of most types of conceptual structure makes it immune
to many of the most telling arguments against its rival positions, but makes it vulnerable
to others.53 Some argue that primitive concepts must be innate, which suggests that the
conceptual atomist position must be radically nativist about concepts.54 This means that
conceptual atomism probably fails to satisfy the acquisition desideratum: postulating that
the ancient Greeks possessed the concepts
CARBURETOR
and
CANADIAN,
for example, is not a
plausible acquisition story.55 Because of its emphasis on direct reference to worldly things,
conceptual atomism also has difficulties with the cognitive content condition, notably with
distinguishing between coextensive concepts, including those that are empty because they
refer to nonexistent entities. That is, it seems that conceptual atomism not only has difficulty distinguishing
SUPERMAN
from CLARK KENT, but may even have trouble telling SUPERMAN
50
Fodor 93.
Laurence and Margolis 60. A lexical concept is one that can be expressed by a single word, such as BIRD
(Prinz 22). This qualification is important because conceptual compositionality requires that non-atomic
compound concepts such as YELLOW BIRD also exist.
52
Prinz 90.
53
Laurence and Margolis 62.
54
Primitive concepts cannot be decomposed into more basic concepts. They are the bedrock that prevents
compositional processes from falling into infinite regress (Prinz 94).
55
Laurence and Margolis 62–3. One might argue that CARBURETOR and CANADIAN are not primitive concepts
(while this seems plausible, it does not seem immediately obvious exactly what primitives they are then
composed of), and that logical atomism only requires primitive concepts to be innate. Fodor has defended
his version of conceptual atomism against objections to nativism but there is no room or need to discuss this
debate here.
51
68
and
LEX LUTHOR
(or, for that matter,
SUPERMAN
and
UNICORN)
apart.56 In particular, this
criticism seems to indicate an incompatibility between conceptual atomism and the belief
that mathematical entities are non-existent or fictional. Furthermore, as we shall see shortly,
CMT posits that even the most elementary concepts possess internal structure and is therefore also incompatible with conceptual atomism. Because of these incompatibilities with key
elements of this dissertation, and the lack of a definitive argument in its favour, I shall discuss
conceptual atomism no further.
What conclusions should be drawn from this discussion of concepts? First, empirical
and philosophical developments over the last hundred years have led many scholars to reject
the classical theory of concepts, principally because of its failure to satisfy the scope and
categorization criteria. Thus, it would seem prudent to avoid importing this understanding
of concepts into metaphor research, or into this dissertation. Note, however, that rejecting
the idea that concepts are constitutively definitional does not mean concluding definitions
have no place in philosophy or conceptual reasoning. Second, none of the leading approaches
in contemporary concept research seem clearly superior to the others; each do an excellent
job of satisfying some of Prinz’s desiderata while failing to satisfy others. It is possible
that concepts may possess multiple kinds of structure and a single comprehensive theory
of the type sought thus far might be impossible.57 As we shall see, the theory of concepts
underpinning CMT is not a clear instance of any one of the major approaches to concepts.
This should not be regarded as sufficient grounds for suspicion toward this view; indeed, this
rather seems to be a boon insofar as it means that CMT is not immediately subject to the
standard objections leveled at the leading theories. Bearing this in mind, I will now move
my attention to the details of CMT.
3.2
Conceptual Metaphor Theory
Metaphors We Live By begins with a claim that is the core premise of conceptual metaphor
theory: “metaphor is typically viewed as characteristic of language alone, a matter of words
56
57
Laurence and Margolis 69.
Laurence and Margolis 72.
69
rather than thought or action. . . We have found, on the contrary, that metaphor is pervasive
in everyday life, not just in language but in thought and action. Our ordinary conceptual
system, in terms of which we both think and act, is fundamentally metaphorical in nature.”58
An additional qualification clearly shows Lakoff and Johnson are directly opposed to at least
some traditional ideas about metaphor. They tell us that “[m]etaphor is primarily a matter
of thought and action and only derivatively a matter of language.”59 Thus, while many
agree with Hobbes that metaphorical language is an in-principle eliminable impediment to
the clear communication of ideas, Lakoff and Johnson propose that metaphors in speech and
writing are, rather, merely symptoms of metaphoricity inherent in the concepts underlying
our language and its use. Developing this notion of conceptual metaphor and exploring its
ramifications is, indeed, the primary objective of Metaphors We Live By; as such, almost
every instance of the word “metaphor” in that text refers to conceptual metaphor and not
to the narrower idea of metaphorical language.60 This is noteworthy because many of the
objections to CMT arguably depend on an equivocation that stems from misinterpreting
Lakoff and Johnson’s claims about conceptual metaphor as pertaining directly to linguistic
metaphors. Conceptual metaphor is intended to explain metaphorical language: the two are
not identical. As Lakoff and Johnson say, “[m]etaphors as linguistic expressions are possible
precisely because there are metaphors in a person’s conceptual system.”61
Lakoff and Johnson, however, do not offer a concise definition of conceptual metaphor at
the beginning of Metaphors We Live By, but instead approach the idea through a series of examples intended to induce understanding. The clearest description of conceptual metaphor is
given twenty years and thousands of pages later in Where Mathematics Comes From: a conceptual metaphor is “a grounded, unidirectional, inference-preserving cross-domain
mapping — a neural mechanism that allows us to use the inferential structure of
one conceptual domain to reason about another.”62 This dense definition surely begs
58
Metaphors We Live By 3. It is thus clear from the very first paragraph that CMT presumes a fairly
specific, non-traditional notion of “concept”; I will however postpone explicit discussion of the way concepts
are understood in CMT for the moment and focus first on the idea of conceptual metaphor.
59
Lakoff and Johnson, Metaphors We Live By 153.
60
Lakoff and Johnson, Metaphors We Live By 6: “whenever in this book we speak of metaphors. . . it should
be understood that metaphor means metaphorical concept.”
61
Lakoff and Johnson, Metaphors We Live By 6.
62
Lakoff and Núñez, Where Mathematics Comes From (New York: Basic Books, 2000), 6; emphasis mine.
70
to have its constituents unpacked.
Whether one understands mapping cartographically, mathematically, or more generally,
it involves positing or representing a structural correspondence between two things. On a
standard road map, for example, some features of the physical environment are represented
by certain markings on a piece of paper. Describing metaphors as mappings seems quite apt,
insofar as maps focus on certain features of the terrain while ignoring others entirely (for
example, topographical maps and population density maps involve very different saliences).
If a map became accurate in every detail and did not mask or omit any of the structure or
features of what it represented, it would no longer be a map but rather a full-fledged identical
copy of the thing represented. As with maps, it seems essential that metaphors involve partial
rather than total mappings: identities are not metaphors. Although Metaphors We Live By
makes no explicit use of mapping language in describing conceptual metaphor, it is consistent
with and even gestures towards such a position in claiming that “[t]he essence of [conceptual]
metaphor is understanding and experiencing one kind of thing in terms of another.”63 The
necessarily partial nature of metaphorical transference, on the other hand, is already explicitly
present: “part of a metaphorical concept does not and cannot fit.”64
Works subsequent to Metaphors We Live By in the CMT canon do explicitly discuss conceptual metaphor using mapping terminology. In “The Contemporary Theory of Metaphor,”
for example, Lakoff says the following about the nature of conceptual metaphors qua mappings:
Mappings should not be thought of as processes, or as algorithms that mechanically take source domain inputs and produce target domain outputs. Each mapping should be seen instead as a fixed pattern of ontological correspondences
across domains that may, or may not, be applied to a source domain knowledge
structure or a source domain lexical item.65
Describing conceptual metaphor in terms of static correspondences which may be activated
rather than real-time algorithmic derivations coheres with the formal set-theoretic definition
of a mapping as a collection of ordered pairs of elements and thus supports understanding
63
Lakoff and Johnson, Metaphors We Live By 5; emphasis theirs.
Lakoff and Johnson, Metaphors We Live By 13.
65
“The Contemporary Theory of Metaphor” 210.
64
71
conceptual metaphors specifically as mathematical mappings.66 In more recent works, Lakoff
and his collaborators have maintained that the claim “conceptual metaphors are mathematical mappings” is itself metaphorical since “mathematical mappings do not create target
entities, while conceptual metaphors often do.”67 Since 1997, however, they have focused
their efforts on the development of a neural theory of language and have accordingly begun
to theorize conceptual metaphors as neural mappings rather than mathematical mappings.68
While there are reasons to appreciate their neural theory of conceptual metaphor, Lakoff and
Johnson’s abandonment of the mathematical-mapping metaphor may have been somewhat
hasty and unfortunate; I address this issue in chapter 5.
If conceptual metaphors are mappings then it seems obvious that they must be mappings
between concepts. However, in the definition Lakoff uses the phrase “conceptual domain,”
a phrase which he does not explicitly define. While it is possible that “conceptual domain”
could be interpreted to simply mean “concept qua domain of a mapping,” I speculate that
Lakoff’s use of this phrase is meant to convey that these mappings are not confined to
isolated concepts but rather that they usually extend to include larger regions of connected
and structured conceptual networks; it also reminds the reader that conceptual metaphors
typically are partial in that they do not exhaustively map their entire input concept to their
output concept, as noted above.69 After their neurological shift, Lakoff and Johnson define
the domains of metaphorical mappings as “highly structured neural ensembles in different
regions of the brain.”70
66
Saunders Mac Lane, Mathematics: Form and Function (New York: Springer-Verlag, 1986), 129. In
most branches of mathematics, “mapping” is typically used as a synonym for “function.” A mathematical
function is a relation between a collection of inputs (the domain of the function) and a collection of outputs
(the codomain) such that each input is related to one output. However, Lakoff’s understanding of mapping
is closer to a partial binary relation than a function as it does not seem to include the constraint that each
input correspond to at most one output. This technical clarification is of little consequence at this point but
becomes more pertinent in chapter 5.
67
Lakoff and Johnson, Metaphors We Live By 252.
68
Lakoff and Johnson, Metaphors We Live By 254.
69
It seems plausible that Lakoff may intend to invoke fellow cognitive linguist Ronald Langacker’s notion of
a cognitive domain: “any sort of conceptualization: a perceptual experience, a concept, a conceptual complex,
an elaborate knowledge system, and so forth” (Langacker, “An Introduction to Cognitive Grammar” 4).
70
Lakoff and Johnson, Metaphors We Live By 256. While Lakoff does ultimately wish to provide a neurobiological account of concepts, he does not seek to eliminate our phenomenological understanding; he describes
himself as a “noneliminative physicalist.” As will be discussed below, CMT allows for multiple, legitimate,
ineliminable levels of explanation; in my estimation, this is one of the theory’s major benefits (Lakoff and
Johnson, Philosophy in the Flesh 108–14).
72
At the end of chapter 2 I argued that symmetry considerations feature among the properties of metaphor that any sufficient account of metaphor must address; this was motivated
by examples of asymmetric linguistic metaphor such as the ubiquitous “That surgeon is a
butcher.” In general, conceptual metaphors are unidirectional: understanding concept X in
terms of concept Y is not the same as understanding concept Y in terms of concept X. Most
conceptual metaphors are oriented in such a manner to allow us to comprehend a less well
grasped concept in terms of a better understood concept. This is not to say that there are no
instances where a distinct conceptual metaphor maps concept Y to concept X, or to rule out
the possibility of genuinely bidirectional mappings between conceptual domains, metaphorical or otherwise. The claim is simply that the typical conceptual metaphor is unidirectional.
This unidirectionality is related to one of conceptual metaphor’s other definitional properties, cross-domainedness. The power of metaphor seems to reside in its ability to connect
disparate domains. Any mapping from some concept to itself does not involve seeing one
domain in terms of another but only seeing some domain on its own terms. Unidirectionality
and cross-domainedness are connected insofar as any mapping whose source domain and target domain are identical is bidirectional in a certain sense, though not necessarily invertible.
It should be noted that, unlike metaphor, some kinds of conceptual mappings or relations —
subcategorization and metonymy, for example — occur within a single domain.
Conceptual metaphor mappings do not involve random correspondences, but rather systematically preserve the overarching structure relating elements within a conceptual domain.
By preserving inferential structure, conceptual metaphors enable a transfer of understanding that allows the comprehension of difficult, abstract, obscure, or alien concepts in
terms of those better known; that is, conceptual metaphor “sanctions the use of source
domain language and inference patterns for target domain concepts.”71 Thus, Lakoff and
Johnson assert that “the preservation of inference is the most salient property of conceptual
metaphors.”72 In more recent work, Lakoff and Johnson distinguish between structural inferences and enacted inferences, both of which can be preserved by conceptual metaphors.
Structural inferences — such as inferring that Bollo is a mammal from the fact that Bollo
71
72
Lakoff, “Contemporary Theory of Metaphor” 208.
Lakoff and Johnson, Philosophy in the Flesh 58.
73
is a gorilla — can arise non-imaginatively from the static structure of a conceptual system.
Enacted inferences occur when reasoning about dynamical processes and involve an imaginative enactment of the process in question — if I tell you “Bollo beat his chest,” you may
actively imagine a gorilla beating his chest and reach conclusions like “a thumping sound
resulted” based on that imaginative project. Much of our reasoning involves an interaction
of both kinds of inferences. It is important to note that in CMT enacted inferences are
embodied — that is, are both allowed for and constrained by the particulars of our biological apparatus and physical experiences of a shared environment — and occur in the source
domain. Thus, even though enacted inferences are dynamic, the correspondences making up
conceptual metaphors need not be dynamic to preserve such inferences — just as fixed rods
may transfer movement from a puppeteer to a puppet.73
It is imperative to CMT that conceptual metaphors are not arbitrary mappings but are
rather grounded in human embodiment and embodied experiences. Lakoff and Johnson
have claimed that “most of our conceptual system is metaphorically structured; that is, most
concepts are partially understood in terms of other concepts.”74 If conceptual metaphors
are arbitrary, this would entail that most concepts are arbitrary, a clearly problematic position unable to satisfy several of Prinz’s desiderata, notably publicity. In Metaphors We
Live By, Lakoff and Johnson tell us that “conceptual metaphors are grounded in correlations
within our experience”75 but caution that they “do not know very much about the [actual]
experiential bases of metaphors”76 and thus their claims about the experiential foundations
of specific conceptual metaphors are fairly speculative. These intuitions were later formulated into the Grounding Hypothesis. This hypothesis proposes that all concepts have at
least some aspects that are semantically autonomous, in the sense that they emerge directly
and non-metaphorically from regularities in our embodied experience, and that conceptual
metaphors are grounded in this semantically autonomous structure.77 While there might
be some parallels between this hypothesis and the traditional idea that linguistic metaphors
73
Lakoff and Johnson, Metaphors We Live By 259–60.
Metaphors We Live By 56.
75
Metaphors We Live By 154–5; emphasis theirs.
76
Lakoff and Johnson, Metaphors We Live By 19.
77
Lakoff and Turner, More Than Cool Reason (Chicago: U of Chicago P, 1989), 113.
74
74
are grounded in underlying literality, Lakoff and Turner explicitly warn the reader that the
Grounding Hypothesis is about concepts and not about language. They also remind us that
it primarily concerns our body-dependent experiences rather than some human-independent
reality and that it is consistent with the above claim that most concepts are not entirely
semantically autonomous but rely on conceptual metaphor for much of their structure.78
Citing experimental results in cognitive science, Lakoff and Johnson now hold that the
Grounding Hypothesis has been confirmed. It is thus worth examining the details of their
account of conceptual grounding. All embodied beings we know of rely on genetically determined discriminatory capacities for their survival. Even organisms such as plants and
single-celled protists can be observed differentiating light from dark, up from down, food
from non-food, threats from non-threats, etc. in their movements and reactions.79 Thus,
because many such discriminatory mechanisms are a hardwired part of human anatomy and
physiology, we preconsciously and inescapably categorize the world in various fundamental
and stable ways as we interact with it. Because many of our earliest and most basic experiences have to do with bodily survival in the physical world, they possess what CMT
calls image-schematic structure, that is, a sparse sensorimotor form relating to our possible
movements and actions. Such image schemata themselves possess an internal logic that allows for our action-guiding reasoning about the physical world.80 Thus the particularities
of human embodiment ensure that our experiences necessarily involve specific kinds of categorization, and non-metaphorical primitive concepts emerge from this foundation. There
78
Lakoff and Turner 119.
Lakoff and Johnson, Philosophy in the Flesh 17.
80
There is not enough room to present a thorough account of the research on image schemata here; more
details can be found in almost any work by Lakoff and/or Johnson published since 1986. However, three key
quotes will bolster the above synopsis:
79
⋄ “An image schema is a recurring, dynamic pattern of our perceptual interactions and motor programs
that gives coherence and structure to our experience” (Johnson, “Body in the Mind” xiv).
⋄ “Image schemata are gestalt structures, consisting of parts standing in relations and organized into
unified wholes, by means of which our experience manifests discernible order” (Johnson, “Body in the
Mind” xix).
⋄ “image schemata are not propositional, in that they are not abstract subject-predicate structures. . . that specify truth conditions or other conditions of satisfaction. . . They exist, rather, in a
continuous, analog fashion in our understanding. . . On the other hand, image schemata are not rich,
concrete images or mental pictures, either” (Johnson, “Body in the Mind” 23).
75
is extensive agreement between individuals’ versions of these primitive concepts due to the
universality of many of our basic embodied experiences, thanks to humans having developed
in almost-identical environments according to almost-identical genetic instructions.81 Thus,
this theory provides an explanation of the universality of primitive concepts without requiring innateness.82 Conceptual metaphors that have non-metaphorical spatiomotor concepts
as their domains are said to be primary metaphors and are directly grounded; other complex
metaphors obtain their grounding indirectly through their associations with such primary
metaphors.83 Much of Lakoff’s recent CMT research is aimed at enriching this picture of
primitive concept development with further neurological detail.84
A substantial portion of the CMT corpus is devoted to discussing the specific conceptual
metaphors that Lakoff and his coauthors believe constitute the core of our conceptual system
and, although the above definition does a good job of introducing the idea of conceptual
metaphor, some specific examples are useful for illustration. The input and output of a
conceptual metaphor are referred to respectively as the source domain and target domain.85
CMT denotes conceptual metaphors as follows:
81
TARGET-DOMAIN IS SOURCE-DOMAIN.
86
Given the
It may already be obvious to some readers that there are deep parallels between certain passages of
Wittgenstein’s writing and parts of CMT. For example, Wittgenstein’s notion of “seeing-as,” as discussed in
Philosophical Investigations IIxi, seems related if not outright identical to the “seeing-in-terms-of” that is a
core aspect of conceptual metaphor: “The essence of metaphor is understanding and experiencing one kind
of thing in terms of another” (Lakoff and Johnson, Metaphors We Live By 5; emphasis theirs). At times,
Wittgenstein seems wholly sympathetic to the cognitive linguistics project: “If the formation of concepts can
be explained by facts of nature, should we not be interested, not in grammar, but rather in that in nature
which is the basis of grammar?” (Philosophical Investigations 230e ). More specifically, the Wittgensteinian
idea that our language games depend upon our specific form of life may be directly analogous to the Lakovian
notion that language is grounded in embodiment: “[human beings] agree in the language they use. That is not
agreement in opinions but in form of life” (Wittgenstein, Philosophical Investigations 88e ). The idea that a
shared human form of life provides the intersubjectivity necessary for linguistic communication also seems to
be reflected in Wittgenstein’s somewhat cryptic statement “If a lion could talk, we could not understand him”
(Philosophical Investigations 223e ). Lakoff and Johnson do briefly mention Wittgensteinian forms of life, but
only hint at the connexions between their work and his, mostly suggesting that he remained fettered by the
assumptions of analytic philosophy despite his positive suggestions (Philosophy in the Flesh 450). Discussing
the relationships between Wittgenstein and Lakoff in detail is beyond the scope of this dissertation.
82
Lakoff and Johnson, Philosophy in the Flesh 57. “Primitive” does not mean free from internal structure,
but only that such concepts have a kind of priority.
83
This distinction will be discussed further below, with explicit examples provided. The interested reader
is directed to pages 50 to 54 of Philosophy in the Flesh, where a list of representative examples of primary
metaphor is provided.
84
See Gallese and Lakoff (2005) and Lakoff (2008) for more details.
85
One must be careful not to confuse these instances of the word “domain” with the technical mathematical
usage described in an earlier footnote.
86
In more recent writings, Lakoff and Johnson sometimes use an alternative notation that will seem more
76
prevalence of traditional linguistic theories of metaphor, it is easy to forget that
SOURCE
TARGET IS
is merely a notational mnemonic and not the metaphor itself; to avoid equivocation,
it is imperative to remember that a conceptual metaphor is the collection of correspondences
between portions of the source domain and the target domain.87 I now turn my attention to
several central examples of conceptual metaphor used by CMT.
The conceptual metaphor AFFECTION IS WARMTH is integral to how we experience and reason
about affection.88 This is evidenced, in part, by the unconscious ease with which we produce
and interpret language involving this metaphor:
⋄ We shared a warm embrace.
⋄ Edith was hot for Darrell. Darrell, too, was burning with passion.
⋄ My advances generated only a lukewarm response.
⋄ She acknowledged him frostily.
⋄ Olaf didn’t stay the night because he is frigid.
⋄ Sally is a block of ice.89
According to CMT, these sentences are all metaphorical expressions of the same underlying
conceptual metaphor; that is, producing or understanding these sentences involves conceptualizing affection in terms of warmth.90 Moreover, it should be noted that all of these
examples are more or less common everyday utterances, not ostentatious poetry: outside of
the context of being presented as examples of metaphorical utterances, their metaphoricity
likely would pass unnoticed. Speaking about affection in terms of warmth is normal and
conventional, not exceptional. Whereas such cases force linguistic theories of metaphor to
provide an account of “dead metaphors,” they cause no such complication for CMT because
natural and familiar to those with mathematical training: Source-Domain→Target-Domain. Despite these
benefits, I will use the small capital letter notation because it is standard in the literature.
87
Lakoff, “Contemporary Theory of Metaphor” 207.
88
Lakoff and Johnson, Metaphors We Live By 255.
89
The last example is borrowed from Searle (“Metaphor” 82).
90
Of course, this charitably assumes that these sentences occur in a typical context, that is, one that
connects them to this metaphor. One should remain cognizant of the contextuality of language throughout
this discussion of conceptual metaphors.
77
understanding a sentence using concepts that are structured by conceptual metaphors does
not require conscious recognition that this is taking place.91
AFFECTION IS WARMTH
is a paradigmatic example of a primary conceptual metaphor. Pri-
mary metaphors “are directly grounded in the everyday experience that links our sensorymotor experience to the domain of our subjective judgments,”92 and, as such, are a foundational element of our conceptual system. Indeed, the universality of the collection of primary
metaphors among humans is what allows me to speak about our conceptual system rather
than my conceptual system. Yet primary metaphors are learned rather than innate: “[w]e
acquire a large system of primary metaphors automatically and unconsciously simply by functioning in the most ordinary of ways in the everyday world from our earliest years.”93 Lakoff
and Johnson speculate that the
AFFECTION IS WARMTH
metaphor is acquired in the following
way. When a parent is affectionate towards an infant, they typically snuggle close to them,
thereby warming the baby with their body heat. These early simultaneous experiences of
warmth and affection lead the infant to strongly associate the two domains.94 In the neural
version of CMT, the simultaneity of early experiences of warmth and affection are claimed to
incite a neural conflation, a neural linkage between the temperature-devoted and emotiondevoted regions of the brain.95 Pithily, Lakoff and Johnson tell us that this conflation arises
because “neurons that fire together wire together.”96 When affection and warmth are later
91
Conceptual metaphor theorists hold that such conventional entrenched conceptual metaphors are thoroughly alive, as they enjoy copious (if unconscious) usage. A dead conceptual metaphor is a cross-domain
mapping that is no longer in common usage. Lakoff’s canonical example involves the English word “pedigree,” which derives from pie de grue, Old French for “crane’s foot.” Whereas the metaphor connecting the
branching of a family tree with the shape of a bird’s foot was once part of the concept of pedigree, this mapping has become etymological trivia rather than an active part of our conceptual system (Lakoff, “Death”
144).
92
Lakoff and Johnson, Metaphors We Live By 255.
93
Lakoff and Johnson, Philosophy in the Flesh 47.
94
While this account of the origins of the AFFECTION IS WARMTH metaphor seems plausible, it may be overly
speculative and simplistic, a frequent criticism leveled against CMT. For example, Lakoff and Johnson do
not discuss the possibility that hormones responsible for our experienced feelings of affection also play a
role in body temperature regulation. There is some experimental evidence that pleasant one-on-one social
interactions between adults are correlated with an increase in core body temperature (Dabbs and Moorer
518). Ultimately, such criticisms target explanations of how specific conceptual metaphors arise and would
require substantial bolstering to upgrade them to a serious attack on CMT generally.
95
Lakoff and Johnson, Philosophy in the Flesh 48. Lakoff and Johnson credit Christopher Johnson — not
to be confused with CMT pioneer Mark Johnson — with developing the theory of neural conflation they
adopt. See C. Johnson (1999) for more details.
96
Lakoff and Johnson, Metaphors We Live By 256.
78
experienced separately, the associations persist, just as a channel remains even when a river is
not actively flowing through it. In this way, hundreds of primary metaphors form automatically and unconsciously in all people under normal developmental circumstances, based upon
the shared aspects of human embodiment; for example,
AFFECTION IS WARMTH
specifically de-
pends strongly on our mammalian biology: we are warmth-radiating, temperature-sensitive
beings that attend closely and lovingly to our young.
AFFECTION IS WARMTH
is unidirectional, allowing us to understand affection in terms of
warmth but not vice versa; for example, nobody would understand that you wanted to turn
up the thermostat if you walked into a room and asked “Can we make the room a bit more
loving?” Primary metaphors are naturally oriented from the sensorimotor to the subjective;
this is because sensorimotor concepts possess more native inferential structure, often in the
form of image schemata, than concepts based in abstract subjective experiences such as
emotions.97 For example, the temperature receptors in my skin constantly provide me with
information about the warmth of my environment that allows me to take action to maintain
my optimal body temperature. I experience warmth as possessing degrees of intensity as well
as a directionality (that is, warmer than me or colder than me). Because these receptors are
spread across my surface, different parts of my body may experience different intensities and
thus I often also experience warmth as emanating from a source. These structural elements
in the resulting concept of
WARMTH
allow me to make inferences like “If I wish to stop being
cold, I should move away from the source of the coldness.”98 When I think about affection,
I metaphorically import inferential structure from my understanding of warmth: “If I don’t
like that she is cold to me, I should distance myself from her.” We do not always understand
affection in terms of warmth; for example, this metaphor may remain inactive while another
metaphor — such as AFFECTION IS SWEETNESS — provides inferential structure.99 Our concept
of affection is largely structured by conceptual metaphors; without them, it only possesses
97
Lakoff and Johnson, Philosophy in the Flesh 55. This portion of Lakoff and Johnson’s account synthesizes
Joseph Grady’s theory of primary metaphor and Srinivas Narayanan’s neural theory of metaphor. See Grady
(1997) and Narayanan (1997) for more details.
98
While we may have some conceptual metaphors with warmth as the target domain, the important point
is that the concept of warmth possesses a significant amount of inherent inferential structure, enough to
directly ground any conceptual metaphor that features it as the source domain.
99
Lakoff and Johnson, Philosophy in the Flesh 56.
79
a minimal skeleton of structure. While we might theoretically be able to perform a small
amount of reasoning based solely upon this literal skeleton of
AFFECTION,
it would take a
considerable, and likely counterproductive, effort to isolate ourselves from the conceptual
metaphors that also make a contribution.100 While primary metaphors — such as AFFECTION
IS WARMTH, MORE IS UP, SIMILARITY IS CLOSENESS, CATEGORIES ARE CONTAINERS,
etc. — form the
foundational ground of our conceptual system, a second large class of conceptual metaphors
are not directly grounded.
Many of our most important conceptual metaphors have non-sensorimotor source domains
with significant inferential structure. Lakoff and Johnson call these complex metaphors.
Some examples include
TIME IS MONEY, PEOPLE ARE PLANTS, A LIFETIME IS A DAY,
A JOURNEY. TIME IS MONEY
and
LOVE IS
is a paragon instance of a complex metaphor, one in which an
abstract concept metaphorically lends its structure to another abstract concept: money is
a complicated human creation, not a basic sensorimotor percept. In our culture, we make
thorough use of this metaphor: we are constantly trying to think of new ways to save time,
of worthwhile ways to invest our time rather than spending it frivolously, and so forth.
We have institutionalized this metaphor in the practice of paying people’s wages based on
amount of time spent working, for example. However, other cultures — such as that of
Pueblo communities — do not seem to conceptualize time as a resource. The absence of this
metaphor from such cultures may be apparent in various ways, from the lack of linguistic
expressions employing monetary language to describe temporal phenomena to a pace of life
drastically different from ours thanks to the absence of time budgeting.101 Another example
of a cultural variation in complex metaphor can be seen in one of the ways the future is
conceptualized. It is common for humans to conceptualize themselves as moving observers
going into a fixed future. However, in some cultures — such as my own — we tend to think of
ourselves as facing the future and moving into it, while other cultures — such as the Aymara
of the Andes — see themselves as facing the past and carefully walking backwards into an
unseen future.102 Thus, while most primary metaphors are universal among humans, there
seems to be room for variation in complex metaphors from culture to culture.
100
Lakoff and Johnson, Philosophy in the Flesh 58–9.
Lakoff and Johnson, Philosophy in the Flesh 164–5.
102
Lakoff and Johnson, Philosophy in the Flesh 141.
101
80
Cultural variations in complex metaphors raise the question of the origins and grounding
of metaphor. Lakoff and Johnson explain that complex metaphors are “built out of primary metaphors plus forms of commonplace knowledge: cultural models, folk theories, or
simply knowledge or beliefs that are widely accepted in a culture.”103 Thus, though complex metaphors may not have direct experiential grounding of their own they are indirectly
grounded by way of their component primary metaphors.104 For example, there is no simple
experiential correlation between purposeful lives and journeys that could ground the complex
metaphor
A PURPOSEFUL LIFE IS A JOURNEY.
Rather, Lakoff and Johnson claim this metaphor
is grounded because it is the synthesis of two primary metaphors —
TIONS
and
ACTIONS ARE MOTIONS
PURPOSES ARE DESTINA-
— combined with the cultural belief that every person has
a purpose in life that they should strive to achieve.105 Syntheses of primary metaphors like
this are conjectured to arise by means of conceptual blending, a notion extensively developed
by Gilles Fauconnier and Mark Turner.106 A conceptual blend is an operation on two input
mental spaces that yields a third blended space that “inherits partial structure from the input
spaces and has emergent structure of its own.”107 The blended space inherits structure from
partial cross-space mappings between counterpart facets of the input spaces;108 the blend
also has emergent structure that arises through the related processes of composition (new
relations forming between elements projected from the input spaces), completion (importing
elements from cultural or other cognitive models and schemata to integrate disparate projected elements into coherent wholes), and elaboration (making conceptual inferences using
103
Philosophy in the Flesh 60.
Lakoff and Johnson, Philosophy in the Flesh 63.
105
Philosophy in the Flesh 61.
106
Conceptual blending theory is an alternative cognitive approach to conceptual structure that shares many
motivations and premises with CMT. While Lakoff suggests that the two theories are “different in scope and
intent” (Lakoff and Johnson, Metaphors We Live By 261), both he and others seem to believe that “the two
approaches are complimentary” (Grady, Oakley, and Coulson 420). For more details on conceptual blending,
see Fauconnier and Turner (2002).
107
Fauconnier 149; emphasis his. Mental spaces are to conceptual blending theory as conceptual domains
are to CMT, the objects that form the domains of the mappings under consideration. Some authors have
suggested that mental spaces are partial, temporary representational structures that depend upon the more
stable and systematic conceptual domains (Grady, Oakley, and Coulson 421). Considering the detailed
differences between these two kinds of mental constructs is a task for another time.
108
In some cases, these partial cross-space mappings are themselves conceptual metaphors (Grady, Oakley,
and Coulson 428).
104
81
the emergent logic of the blend).109 Lakoff and Johnson do not discuss the technical details
of metaphorical blending, and neither will I. The important point here is that the myriad
complex metaphors that structure our conceptual systems and cultural institutions are not
problematically arbitrary but are grounded indirectly through associated primary metaphors
and the semantically autonomous aspects of their domains; moreover, plausible suggestions
have been made regarding mechanisms of complex metaphor formation.
If many primary metaphors are universal among humans, and complex metaphors are
syntheses of primary metaphors, how can we explain the cultural variation observed in the
structuring of fundamental concepts? The answer is that although there are extensive universal regularities in the embodied experience of humans, peoples of different cultures also have
differing experiences based in the peculiarities of their local environment and their accidental
history.110 Cultural beliefs and biases can enter into complex metaphors as the “commonplace knowledge” component mentioned above. Conceptual metaphors, whether primary or
complex, are grounded in experiential bases. It does not follow from this that having an experiential basis necessarily entails possession of an associated conceptual metaphor. As Lakoff
puts it, “[e]xperiential bases motivate metaphors, they do not predict them. Thus, not every
language has a
between
MORE
MORE IS UP
metaphor, though all human beings experience a correspondence
and UP. What this experiential basis does predict is that no language will have
the opposite metaphor LESS IS UP.”111 Exactly what causes a conceptual metaphor to coalesce
out of an experiential basis in some cases but not others is not explained in the Lakovian
corpus.
Conceptual metaphors, whether primary or complex, emerge from and are grounded in
109
Fauconnier 149–51. Note that the process of completion posited by the conceptual blending theorists
seems directly analogous to — if not outright identical to — how we import or create details to fill in the
gaps between the relatively sparse descriptions presented in novels and other fictions.
110
Lakoff has been criticized for not adequately defining key terms in his theory, including “culture.” I will
discuss this criticism later in this chapter. I understand cultures in a fairly broad sense, as groups of people
united through shared experiences, beliefs, values, etc.; thus, I am a member of many different cultures, of
various cardinalities, some as small as 2.
111
“Contemporary Theory of Metaphor” 241. Note that this is a probabilistically motivated, empirically
testable prediction about the conceptual metaphors grounding the various languages spoken by humans. It
is not impossible that humans could possess the metaphor LESS IS UP, merely unlikely. I leave it up to cultural
linguists and anthropologists to test this prediction. It may be worth noting that the quantity being tracked
seems to make some difference here: a person who lacks sensibility is thought of as having their head in the
clouds, for example. It would be interesting to hear what Lakoff would say about such examples.
82
experiential bases. Conceptual metaphors are also instantiated or realized in the world by
humans. The most obvious realizations are linguistic, sentences that depend upon underlying
conceptual metaphors for their construction and interpretation. In addition, conceptual
metaphors are instantiated throughout the spectrum of human creations: in artistic works, in
instruments and tools, in buildings, in social practices and observances, and in other cultural
artifacts.112 For example, the architecture of cathedrals relies upon metaphors like
UP,
GOOD IS
with lofty cavernous arched ceilings being conceived as better tributes to the divine than
squat huts. Historical timelines depend upon the metaphor
EXTENTION.
Cartoons often invoke conceptual metaphors like
TEMPORAL EXTENTION IS SPATIAL
ANGER IS HEAT
visually: angry
characters routinely transform into whistling kettles or soaring thermometers.113 Importantly,
Lakoff notes that “[e]xperiential bases and realizations of metaphors are two sides of the
same coin: they are both correlations in real experience that have the same structure as the
correlations in metaphors. The difference is that experiential bases precede, ground, and make
sense of conventional metaphorical mappings, whereas realizations follow, and are made sense
of, via the conventional metaphors. And, as we noted above, one generation’s realizations of
a metaphor can become part of the next generation’s experiential basis for that metaphor.”114
Thus, conceptual metaphors can become dynamically entrenched, originating in experiences
of nature alone but later fortified by human-created instantiations of the metaphor within
the shared environment. An analogous example is pertinent to the next chapter. There are
few rectangles, circles, and regular polygons in natural environments untouched by human
manipulation; the faces of certain crystals, the hexagonal cells of a honeycomb, and the
circular visages of the moon and sun are among the short list of examples. However, some
early humans gleaned these shapes from sparse natural sources such as those mentioned
above, and human artifacts now frequently involve such shapes, such as the rectangular page
you are currently reading. Indeed, rectangles and circles are now ubiquitous to the point of
our experience being permeated with them, so much so that these shapes are among the first
things children are formally educated about. It is one of the wonders of animal mimicry that
112
Thus, the conceptual metaphors possessed by some culture, institution, or discipline are often constitutive
of its identity.
113
Lakoff, “Contemporary Theory of Metaphor” 241.
114
“Contemporary Theory of Metaphor” 244.
83
we humans are able to reproduce and thereby amplify various elements of our experience,
including conceptual metaphors.
A third kind of conceptual metaphor is discussed in several works in the CMT canon, but
is conspicuously absent from Philosophy in the Flesh and other recent writings. In More Than
Cool Reason, Lakoff and Mark Turner suggest that some metaphors involve correspondences
between relatively specific mental images rather than between conceptual domains.115 Unlike
other conceptual metaphors which underlie multiple expressions in language and culture,
image metaphors are one-shot mappings because the “proliferation of detail in the images
limits image mappings to highly specific cases.”116 For example, Kendall Walton’s example
“Crotone is on the arch of the Italian boot” seems to be underpinned by an image metaphor
that maps a mental image of a boot onto a mental image of the shape of Italy.117 This
metaphor allows me to infer from the above sentence that Crotone is in the southeast of
Italy. Due to their one-shot nature, image metaphors do not tend to play a deep structuring
role in our conceptual system and are often merely aesthetically pleasing or communicative
shortcuts rather than fundamental ways of thinking; for example, the sentences “Crotone is in
southeastern Italy, on the northeast coast of the Calabrian peninsula” and “Crotone is located
at 39◦ 05′ N 17◦ 07′ E” also say where Crotone is located without employing the superficial
boot metaphor. Lakoff and Johnson have always been primarily interested in developing a
comprehensive theory of embodied reason and image metaphors are only peripheral to this
project; they seem to not be metaphors we live by but rather idiosyncratic metaphors that
“stand alone and are not used systematically in our language or thought.”118 It is clear that
many mathematical image metaphors are among the class of extraneous metaphors mentioned
briefly in Where Mathematics Comes From, metaphors that “can be eliminated without any
substantive change in the conceptual structure of mathematics.”119 Canonical examples of
extraneous mathematical metaphors include functions that are named after the shape of
115
Lakoff and Turner 89. Note that the “images” in question here may come from any experiential modality;
they need not be visual in nature, though in practice they often are.
116
Lakoff, “Contemporary Theory of Metaphor” 230.
117
Kendall Walton, “Metaphor and Prop Oriented Make-Believe,” European Journal of Philosophy 1.1
(1993): 40.
118
Metaphors We Live By 54.
119
Lakoff and Núñez, Where Mathematics Comes From 53.
84
their graph, such as the step function (so called because its graph resembles a staircase) and
cardioid (which vaguely resembles a heart). These descriptive names are based on superficial
structural resemblances and are in no way constitutive of the mathematical objects in question
or how we reason about them. Image metaphors are an important part of CMT qua theory
of metaphor and may be progenitors of more persistent conceptual metaphors, but they do
not play a significant role in structuring our conceptual system and are thus absent from
many works of the CMT corpus.
The majority of the metaphors considered by CMT are conventional conceptual metaphors
that are deeply entrenched in the way humans think. Such metaphors are frequently realized in our speech and writing, and typically are invoked without conscious recognition of
their metaphoricity. Most earlier theories at most acknowledge these utterances as “dead
metaphors” and give them little mention, focusing instead on more ostentatiously creative
metaphors, whereas one of the fundamental insights of CMT is that it is reality as we experience it — and not just poetry — that is, in an important sense, metaphorical. Though
this powerful insight regarding conventionalized “metaphors we live by” is its primary focus,
CMT also delivers an account of novel metaphor. This is an important inclusion in the theory
for at least two reasons. First, if CMT is to be seen as a legitimate alternative to existing
philosophical theories of metaphor rather than a parallel theory that misses the point, it
must say something about the phenomena that most other theories take as paradigmatic.
Second, it would seem that even the most conventional conceptual metaphor must have once
been new insofar as CMT holds conceptual metaphors to be acquired rather than innate.
Thus, there is room for at least two kinds of novelty in the Lakovian account of metaphor:
linguistic novelty and conceptual novelty.
First, consider linguistic novelty. Existing conventional metaphors may give rise to genuinely novel linguistic utterances. Through careful, clever construction, a master wordsmith
may breathe new vitality into any idea, even one arising from the most mundane and ubiquitous conceptual metaphor. Such novel realizations may well involve extending the conceptual
mapping to other elements of the source domain in a natural yet heretofore unexploited way.
Consider the following example: the conceptual metaphor
PEOPLE ARE MACHINES
has been
realized for many years in sentences such as “At the end of a ten-hour shift, I’m running on
85
fumes” and “I feel like I’m about a quart low.” A novel metaphorical sentence that realizes
unexploited aspects of the
MACHINES
concept is “The kids these days seem to be running a
new operating system that is incompatible with my aging hardware.” Novel extensions often
reflect conceptual changes in the source domain arising from influences such as the introduction of new technologies; for example, the sentences above reflect the transition from an
age where machines were exclusively mechanical to one where machines have a significant
number of sophisticated electronic components. Under CMT, novel figurative utterances of
this type cannot count as cases of novel conceptual metaphor because the underlying conceptual mappings have been previously established and conventionalized; the best they can do
is blow some dust off of well-worn conceptual metaphors and draw new attention to them.120
For Lakoff, novel conceptual metaphors are new cognitive mappings that arise through
human creativity and imagination.121 It is important to note that “novel” opposes “conventional” in CMT and is thus a technical term that extends beyond mere chronology. Because
conventional metaphors are usually utilized unconsciously, automatically, and effortlessly,
it seems plausible that a use of conscious effort would be a defining characteristic of novel
metaphor.122 A few other features of this understanding of novel metaphor deserve mention.
Novel conceptual metaphors may, but need not, lead directly to a novel linguistic realization;
the metaphor may be implemented experientially without being realized expressively. It is
also plausible that, rarely, an accidentally or randomly generated novel utterance may induce a new conceptual metaphor by serving as an experiential basis, not entirely unlike when
someone makes an unintended pun that is only later recognized as such. It is an important
feature of the way concepts connect to language — including the language of mathematics
— that sentences may be created which satisfy the formal syntactic rules but whose conceptual significance or meaning is only later appreciated.123 Regardless of how they arise, novel
120
One may argue that substantial extension to a conceptual metaphor seems more akin to genuine novelty
than it does to minimal, straightforward extension. However, my purpose here was merely to reemphasize
the distinction between linguistic and conceptual metaphor and to note that linguistic novelty may, but need
not, be accompanied by conceptual novelty. Therefore, I will not address this argument.
121
Lakoff and Johnson, Metaphors We Live By 139.
122
Lakoff, “Contemporary Theory of Metaphor” 245. The CMT understanding of novelty versus conventionality emphasizes the individual rather than the population; it is important to remain cognizant of the
context in which these adjectives are being applied in order to avoid error.
123
A related, more significant case has been discussed briefly above: a speaker may utter a sentence invoking
a conceptual metaphor that their audience does not possess. Such a sentence would be an instantiation of a
86
conceptual metaphors are typically fleeting. Some are adopted into the collective conceptual
system, having their gleam of novelty worn to a patina of conventionality. Others vanish
into oblivion, lacking sufficient interest or underlying conceptual support to persist.124 In
theory, a conceptual metaphor may be compelling and useful, and yet strange enough to defy
conventionalization, thereby remaining novel for an extended duration.
This raises the question of whether all conventional metaphors begin as novel metaphors
in the sense given in the previous paragraph. It is clear that every conceptual metaphor
must have been new at some point: I possessed no conceptual metaphors before I was born,
and there were no conceptual metaphors at all before life came into existence. However,
if novelty is understood to involve conscious creativity then some conceptual metaphors
can never have been novel in this sense. In particular, most primary metaphors cannot
be novel insofar as they arise in humans in infancy, arguably prior to the development of
consciousness.125 On the other hand, the typically one-off nature of image metaphors suggests
that they will frequently be novel, though there are many cases where image metaphors have
been conventionalized and entered our lexicon, such as “computer mouse.” It is difficult
to do more than speculate about whether particular conventionalized complex metaphors
started their lives as novel metaphors or were “metaphors stillborn.”126 For example, while
it seems clear that
LIGHT IS A WAVE
originated as a novel metaphor created by physicists
in the seventeenth century, it is not obvious if
ARGUMENT IS WAR
was ever novel.127 All of
this suggests that there are at least two ways conceptual metaphors may be born: either
they are the conscious, intentional, and relatively abrupt product of an individual, or they
metaphor for the speaker and, potentially, part of the experiential basis of a novel (to them) metaphor for
the audience.
124
If the human brain is constantly seeking to form new connexions, it is plausible that the vast majority
of novel conceptual metaphors are untenable and are abandoned even before they reach the conscious mind.
Random, non-systematic juxtapositions of concepts — like FURNITURE IS CHEESE — are typically entirely
untenable — or at least practically useless. Such speculative hypotheses must be tested by neuropsychological
experimentation that is beyond the scope of this dissertation.
125
Lakoff and Johnson, Metaphors We Live By, 256: “Primary metaphors arise spontaneously and automatically without our being aware of them.”
126
Quine 160.
127
While it is desirable to use original examples in academic work whenever possible, it is hopefully clear
why I am relying on canonical conceptual metaphors here. It would take an unreasonable amount of time and
effort to uncover a common conventional metaphor that Lakoff and his cohort have not already catalogued,
and creating a novel conceptual metaphor that is genuinely new, tenable, and that serves as a good illustrative
example would require a stroke of creative genius that has not yet struck.
87
develop gradually, surreptitiously, and communally and are conventional from conception.128
The idea that much of conceptual development is unconscious and gradual while some is
conscious and abrupt seems relatively uncontroversial yet has important consequences for
any concept acquisition story. One important and controversial claim made by CMT is that
new conceptual metaphors may actually create similarities between concepts.
Chapter 2 looked at popular comparison theories of metaphor which hold that metaphors
are elliptical similes, comparisons of the salient similarities between two things. Comparisons cannot create similarities, they can only pick out pre-existing, isolated resemblances.
In CMT, mainly image metaphors function in this way. Other conceptual metaphors are “typically based on cross-domain correlations in our experience, which give rise to the perceived
similarities between the two domains within the metaphor.”129 Thus, while two things may
superficially resemble each other, any conceptual metaphor connecting them will be based
on deeper experiential correlations. The creation of similarity is governed by Lakoff’s hypothesized Invariance Principle: “Metaphorical mappings preserve the cognitive topology
(that is, the image-schema structure) of the source domain, in a way consistent with the
inherent structure of the target domain.”130 The basic idea is this: concepts have varying
amounts of inherent structure emerging directly from experience, typically in the form of
image schemata. The posited inference preserving nature of conceptual metaphors requires
that if an image schema is mapped from the source domain then it must be aligned with the
structure of the target domain; the inherent structure of the target domain constrains the
mapping possibilities. However, it was above noted that conceptual metaphors tend to be
directed from the less abstract to the more abstract. Abstract concepts such as LOVE possess
relatively little inherent structure. This has two important, related consequences. First, this
lack of structure leaves us with few resources for understanding and reasoning about our
many and varied experiences of love; the inherent structure of love seems to include only an
128
The relationship between novelty and conventionality, particularly with respect to transitions between
the two as well as considerations of conceptual metaphors in individuals versus populations, is one place that
CMT could use some improvement. Cornelia Müller makes some positive suggestions in this direction which
are discussed below.
129
Lakoff and Johnson, Metaphors We Live By 245.
130
“Contemporary Theory of Metaphor,” 215. It should be noted that in the 2003 afterword to Metaphors
We Live By, Lakoff describes the Invariance Principle as “ugly” and “unfortunate,” and cites this as motivation for moving to a neural theory of language (254).
88
impoverished skeleton of a concept: “a lover, a beloved, feelings of love, and a relationship,
which has an onset and often an end point.”131 Second, their lack of structure means that
abstract concepts have fewer limitations arising from the Invariance Principle. This absence
of inherent structure can be seen as a potential for taking on new structure via conceptual
metaphor.132 These two consequences both motivate and allow for the metaphoric importation of structure from less-abstract concepts to aid our understanding. This importation is
not unlike filling in details gleaned from the world on a blank section of a terrain map. It is
this imposition of new structure, the creation of target entities where there once was void,
that underlies the creation of similarity claim.133 The controversy surrounding the creation
of similarity claim will be discussed in further detail below. A third important consequence
of the relative lack of inherent structure is that a single abstract concept can serve as the
target domain for multiple conceptual metaphors.
The less inherent structure a concept possesses, the greater potential it has for becoming
structured via conceptual metaphor. As mentioned above,
LOVE
has very little inherent
structure and so people’s many and varied experiences of love tend to be understood using
conceptual metaphors. Because experiences of love can and do vary drastically, a variety of
metaphors are employed, including
and
LOVE IS WAR.
134
LOVE IS A JOURNEY, LOVE IS NOURISHMENT, LOVE IS MADNESS,
Each of these metaphors emphasizes and explains particular aspects
of love experiences while masking others.
LOVE IS WAR,
for example, provides structure for
understanding experiences of love involving conflict and injury but does little to explain
more cooperative aspects. Thus, abstract concepts will often be structured by a patchwork
of several conceptual metaphors:
Each mapping is rather limited: a small conceptual structure in a source domain
mapped onto an equally small conceptual structure in the target domain. For a
rich and important domain of experience like love, a single conceptual mapping
does not do the job of allowing us to reason and talk about the experience of love
as a whole. More than one metaphorical mapping is needed.135
131
Lakoff and Johnson, Philosophy in the Flesh 70.
It should be noted that even concepts displaying thorough inherent structuring may take on further
structure by way of conceptual metaphor.
133
Lakoff and Johnson, Metaphors We Live By 253. It is worth noting the connexion between the creation
of similarity and the discussion of novel linguistic metaphors extending conceptual metaphors above.
134
Lakoff and Johnson, Metaphors We Live By 141.
135
Lakoff and Johnson, Philosophy in the Flesh 71.
132
89
There are several ways metaphors with a shared target domain may relate to one another.
If the experiences associated with a certain concept are sufficiently diverse, incompatible
or even outright contradictory metaphors may provide structure to the concept. When we
consider that incompatible attributes — such as evenness and oddness — frequently belong
to the same concept, the idea of incompatible metaphors structuring a concept becomes less
objectionable; as long as they are not invoked simultaneously, there is no problem. However,
when incompatible conceptual metaphors are invoked within a single thought or sentence, an
undesirable dissonance arises that explains the traditional prohibition against mixing one’s
metaphors.136 Not all conceptual metaphors are incompatible, and CMT explains that it can
be advantageous to layer or combine metaphors. For example, we have already seen how primary metaphors can positively interact to constitutively ground a complex metaphor. Lakoff
and Johnson sometimes refer to such strongly interacting primary metaphors as consistent.
By contrast, metaphors that positively interact but whose structural overlap is less complete and integrated are said to be coherent. Metaphoric coherences will typically involve the
source domains having common or connected image-schematic structure.137 For example, the
metaphors AN ARGUMENT IS A JOURNEY and AN ARGUMENT IS A CONTAINER may be used constructively in the same sentence to describe our experience of an argument as having a direction
and a content; neither of these metaphors alone captures both of these aspects. One of the
reasons these metaphors play well together is that the PATH schema and the CONTAINER schema
are spatial schemata that can be coherently superimposed: we may follow a path through
a contained space, for example. Such superimpositions can lead to shared entailments, as
illustrated by the sentence “Now that he’s halfway through his argument, I can see where he’s
going.”138 Though Lakoff seems to have abandoned the consistency versus coherency terminology early on, the idea that metaphors can interact in different ways remains legitimate
and useful; in particular, we shall see in chapter 4 that Lakoff and Núñez claim that four
interacting coherent metaphors collectively serve as the foundational ground of arithmetic.139
The idea that metaphorical mappings work in concert to help bring conceptual structure to
136
For an extended, positive discussion of mixed metaphors, see Müller, chapter 5.
Lakoff and Johnson, Metaphors We Live By 95.
138
Lakoff and Johnson, Metaphors We Live By 92, 103.
139
See Where Mathematics Comes From, chapter 3.
137
90
our experiences naturally leads us to question if other cognitive mechanisms help form the
orchestra, as it were.
Linguists and rhetoricians have traditionally distinguished a wide variety of linguistic
tropes — figures of speech or, to be more etymologically faithful, turns of phrase: irony, simile, metaphor, hyperbole, meiosis, metonymy, synecdoche, litotes, alliteration, and tmesis, to
name but a few.140 Many tropes, such as alliteration, bear little resemblance to metaphor
and seem more purely linguistic rather than constitutively conceptual, being typically used
to embellish, enliven, or emphasize. Aristotle already noted this difference between kinds of
tropes in his Poetics.141 However, as mentioned in chapter 2, simile, metonymy, synecdoche,
and analogy do seem relevantly similar to linguistic metaphor and therefore warrant consideration, as a way of clarifying the idea of metaphor by exploring the details of its boundary.
Lakoff seems to endorse such an approach when he says “[t]o understand what is metaphorical, we must begin with what is not metaphorical. In brief, to the extent that a concept is
understood and structured on its own terms — without making use of structure imported
from a completely different conceptual domain — we will say that it is not metaphorical.”142
Lakoff and Johnson have posited conceptual metaphor as a feature of cognition that explains
a variety of phenomena, linguistic and otherwise. The question thus arises, does conceptual
metaphor explain these apparently related linguistic tropes or are they each associated with
distinct conceptual mappings?
Metonymy is a less familiar trope than metaphor; many people will have never heard of
it even though they employ it. Those who are acquainted with the term likely describe it
as a figure of speech wherein one word or phrase is substituted for another based upon an
association between the two; the association involved is usually one of contiguity, with the
substituted term being a property of, a symbol for, or elsehow connected to the replaced
140
While most of these tropes will be familiar or explicitly discussed below, some are less well known.
Meiosis is conspicuous understatement; for example, referring to World War II as ‘a skirmish.’ Notably,
A.P. Martinich claims that several canonical examples of linguistic metaphor — such as “No man is an
island” — are actually instances of meiosis (“A Theory for Metaphor” 454–5). Litotes involves a sort of
double negation, as in endorsing someone’s position by telling her “You’re not wrong.” Tmesis occurs when
a word is bifurcated, frequently so that another word may be inserted as an infix, as in “unbloodylikely” and
“absofuckinglutely.”
141
Poetics, 1457b.
142
Lakoff and Turner 57.
91
term.143 An example of a linguistic metonymy is “I enjoy reading Plato,” where “Plato”
stands for “the writings of Plato.” From its inception, CMT has held a view of metonymy
similar to its view of metaphor: metonymic and synecdochic sentences are merely symptoms
of underlying conceptual metonymies.144 For example, the example above employs the conceptual metonymy
CREATOR FOR CREATION.
However, conceptual metonymy is a distinct kind
of cognitive mapping from conceptual metaphor.145 Whereas metaphor is a cross-domain inferential mapping, metonymy is an intradomain referential mapping: “in a metonymy, there
is only one domain: the immediate subject matter. There is only one mapping: typically
the metonymic source maps to the metonymic target (the referent) so that one item in the
domain can stand for the other.”146 Metaphor is conceived as necessarily involving the crossing of a gap between the source domain and target domain, the use of one thing in thinking
about another, distinct thing. By contrast, metonymy requires that such a gap necessarily
not exist, that the source entity and target entity be connected and “form a single, complex
subject matter.”147 Thus, conceptual metaphor and metonymy must be distinct mechanisms,
as one necessarily involves bridging a gap that is essentially absent in the other: “metonymy
[is] a stand-for relationship between two elements within a single conceptual domain and
metaphor [is] an is-understood-as relationship between two conceptually distant domains.”148
A metonymy can map between seemingly distinct domains, such as TIME and SPACE, as long as
it exploits a pre-existing connexion that yields a single complex subject matter. For example,
conventionalized, contextual understandings of rates of travel allow for the TIME FOR DISTANCE
metonymy underlying the sentence “Regina is two-and-a-half hours from Saskatoon.” Notice
that in this metonymy there is no transfer of logic or structure between the two domains as
there is in the
TIME IS SPACE
metaphor: it is only the nature of the pre-existing relationship
143
“metonymy, n.,” OED Online, Sept. 2011, Oxford UP, 22 Sept. 2011.
In many accounts of figurative language, synecdoche is taken to be a special instance of metonymy. For
CMT, linguistic synecdoches are symptoms of the conceptual metonymy THE PART FOR THE WHOLE (Lakoff and
Johnson, Metaphors We Live By 36).
145
Though they are distinct kinds of mapping, metonymy and metaphor frequently interact in the creation and interpretation of figurative sentences. Moreover, it has been plausibly suggested that conceptual
metaphors may sometimes be motivated by or derived from conceptual metonymies (Kövecses 157).
146
Lakoff and Johnson, Metaphors We Live By 265; emphasis theirs.
147
Lakoff and Johnson, Metaphors We Live By 267.
148
Kövecses 227; emphasis his.
144
92
connecting the two domains that is of interest.149 CMT holds that, like metaphor, “metonymy
is one of the basic characteristics of cognition.”150 Lakoff claims that prototype effects are
often the result of metonymic mappings that substitute either a category member or a subcategory for an entire category; CMT uses conceptual metonymy to explain intraconceptual
graded structure and privileged category elements, including prototypes, stereotypes, ideals,
paragons, and generators.151 Indeed, combining the commonplace and CMT understandings
of metonymy suggests that symbolic thought and symbol use in general — including our linguistic capabilities — may involve conceptual metonymy insofar as we substitute symbols for
the concepts they are associated with; Lakoff and Turner note the existence of a WORDS STAND
FOR THE CONCEPTS THEY EXPRESS
metonymy, though their discussion is dismayingly brief given
the important implications of this mapping.152 While conceptual metonymy is required to
explain both linguistic metonymies and much of our conceptual lives, there is no such thing
as conceptual simile.
While conceptual metaphor cannot explain linguistic metonymy or synecdoche, it does
seem to be integral to the creation and interpretation of many linguistic similes. As mentioned
in chapter 2, similes are usually understood as figurative sentences containing comparisons
that use either “like” or “as.” Identifying such sentences as “figurative” indicates an essential
dissimilarity between the things being compared. That is, most people would be disinclined
to classify comparisons such as “Sex is like sex,” “My hand is like your hand,” or “A kilt is
like a skirt” as similes, whereas two prototypical examples of simile are “She is as slow as
a tortoise” and “My car is like a member of the family.” Leaving the “like” out of the last
example transforms it into a linguistic metaphor: “My car is a member of the family.” This
close connexion between the two tropes led to the comparison theory of metaphor and its
central belief that all metaphors are elliptical similes. Lakoff and Johnson argue that CMT
is incompatible with the comparison theory of metaphor because its understanding of the
similarities involved in simile as isolated and preexisting is in conflict with the metaphoric
149
Lakoff and Johnson, Metaphors We Live By 266.
Lakoff, Women, Fire, and Dangerous Things 77.
151
Women, Fire, and Dangerous Things 90.
152
Lakoff and Turner 108.
150
93
systematicity and the creation of similarity that are integral to their position.153 While some
metaphors (in particular, image metaphors) may merely highlight preexisting perceptual similarities, it seems necessary that something more is going on in those involving more abstract
domains; for example, the interpretation of “I need a moment to digest what you’ve said”
seems to require structural inferences that go beyond the necessarily limited perceptual resemblances between concrete foods and abstract thoughts.154 This argument is apparently
the only place Lakoff explicitly discusses simile but, conveniently, the absence of discussion
itself serves as evidence that simile is not of central concern to CMT. In particular, the entire
lack of discussion implies that there is neither evidence nor theoretical need for “conceptual
simile.” Since metaphor is not reducible to simile in CMT, and there is no distinct cognitive mapping underlying simile, this suggests that similes may be explained by conceptual
metaphor, a suggestion that is supported by the fact that simile often involves understanding
one thing in terms of another.155 Whereas attempted definitions of linguistic metaphor tend
to be problematic and controversial, the definition of simile in terms of its linguistic construction is straightforward, clear, and widely accepted; metaphorical sentences can come in an
infinity of different varieties while similes are more limited in form. While it may be true that
similes are often best accounted for by conceptual metaphor, the details of the explanation
still need filling in, including a discussion of the ramifications of the explicit nature of simile
and the apparently vacuous truth and bidirectionality of “like” statements; further research
in these directions and others may expose simile as a richer and more complex and diverse
phenomenon than is generally believed. As these details are entirely tangential to the present
line of inquiry, I will pursue this no further.
The above paragraph raises a final important question: why does Lakoff refer to crossdomain cognitive mappings as metaphors? The argument in the previous paragraph explains
why it makes sense to refer to the cognitive mappings underlying both metaphor and simile as conceptual metaphors rather than conceptual similes. However, more should be said
about why these maps are called metaphors at all. The terminology originally arises from the
observation that clusters of systematically related metaphorical sentences can be understood
153
Metaphors We Live By 153.
Kövecses 69.
155
Lakoff and Turner 57.
154
94
as symptoms of an underlying cognitive structure. Referring to the underlying cause as the
metaphor makes sense in the same way that referring to a virus as measles does. By defining
metaphor as a kind of cognitive mapping rather than a linguistic device, CMT sidesteps or
dissolves several difficulties that vex traditional views such as the identification problem. In
particular, CMT seems well equipped to handle metametaphors, that is, metaphors used to
comprehend the abstract concept METAPHOR. While the common — and arguably unavoidable
— practice of using metaphor to understand metaphor is problematic on many traditional
accounts, it counts as a point of evidence in favour of CMT: “Every scientific theory is constructed by scientists — human beings who necessarily use the tools of the human mind. One
of those tools is conceptual metaphor. When the scientific subject matter is metaphor itself,
it should be no surprise that such an enterprise has to make use of metaphor.”156 On the other
hand, the fact that “contemporary metaphor theorists commonly use the term ‘metaphor’ to
refer to the conceptual mapping”157 has led to confusion and disagreement insofar as many
people still think of metaphor as a primarily or exclusively linguistic phenomenon. A certain
amount of controversy and criticism might have been avoided by adopting an alternative term
that did not lend itself to unfortunate equivocation.158 For example, to at least some degree,
“conceptual metaphor” seems synonymous with “analogical thought.”159 As the practice of
referring to cross-domain cognitive mappings as (conceptual) metaphors is now over 30 years
old and well established, I will adopt this terminology but try to use it carefully to avoid
equivocating misreadings. However, I am entirely interested in the cognitive mechanism itself
and not particularly concerned about how it is labeled: a cognitive metaphor by any other
name would work as sweetly.
To sum up, Lakoff and his collaborators hold that conceptual metaphors are cognitive
mechanisms that make a fundamental contribution to the human experience. They are neural
mappings between conceptual domains that structure the abstract by preserving inferences
mapped from an ubiquitous and basic sensorimotor schematic ground. Linguistic metaphors
156
Lakoff and Johnson, Metaphors We Live By 252.
Lakoff, “Contemporary Theory of Metaphor” 209.
158
Mathew S. McGlone, “What is the explanatory value of a conceptual metaphor?,” Language and Communication 27.2 (2007): 111.
159
See, for example, Douglas Hofstadter’s extensive writings on analogy.
157
95
are just one of many symptoms of the systematic network of conceptual metaphors — a network that is also utilized in our thought and the physical creations and institutions arising
therefrom. The idea that metaphors are a matter of thought rather than language is not
as novel as Lakoff’s writings sometimes intimate; however, the advent of making conceptual
metaphor the central hub of an ongoing programmatic inquiry into the nature of language
and thought was a revolutionary move that helped his view achieve more success than its
predecessors. CMT has several major strengths. By understanding metaphor as primarily
conceptual and only derivatively linguistic, it bypasses many of the problems plaguing traditional linguistic views that were discussed in chapter 2. CMT’s commitment to empirical
responsibility means that scientific evidence permeates the theory and lends strength to the
supporting arguments. CMT is both parsimonious and fecund, in that conceptual metaphor
and a few other cognitive mechanisms do an enormous — and ever-increasing — amount
of explanatory work. Despite its various strengths, the Lakovian position has many objectors and has not yet found widespread acceptance in the philosophy of language, nor in the
academy generally.
3.3
Criticisms and Objections
Distilling thirty years of evolving CMT canon into a précis is a difficult task. Definitively
addressing the voluminous and diverse body of criticism and debate surrounding this controversial theory seems an insurmountable one. Rather than attempt to consider every criticism
of CMT, the final section of this chapter will focus on a handful of objections that are pervasive in the literature or that are otherwise significant. It is obvious that conceptual metaphor
theory has serious implications for how one understands concepts. Prinz’s desiderata will be
used to expose strengths and weaknesses in this understanding, and thereby fend off misguided objections while identifying others as potentially more telling.160 Further structure
for this section of the chapter comes from Leezenberg’s section on Lakoff’s theory in his
160
It should be noted that a commitment to conceptual metaphor does not necessarily exhaustively determine a theory of concepts; that is, more than one theory of concepts may be compatible with conceptual
metaphor. Thus, even if the particular understanding of concepts discussed below turns out to be fatally
flawed, this does not necessarily require one to abandon the notion of conceptual metaphor.
96
Contexts of Metaphor, a rogue’s gallery of noteworthy objections. I will explain why several
of common philosophical objections to CMT are misguided, and suggest ways that the theory may develop to overcome some of the more telling criticisms. It is worth remembering
at this point that the main reason for this defense of CMT is to allow discussion of Lakoff
and Núñez’s theory of mathematics in chapter 4 to be relevant. Thus, it is not essential
that we commit to CMT wholesale, only that we allow that metaphor is not a purely linguistic phenomenon and that at least some mathematical concepts could be constitutively
conceptualized metaphorically.
Before moving on to the philosophical objections, it should be noted that several authors
have criticized Lakoff’s scholarship. Though many of the books in the CMT canon are
explicitly written for a nonspecialist audience, the near complete absence of in-text citations
is troubling; Jackendoff and Aaron observe that “[t]his compromise between the needs of
informal exposition and scholarly discourse strikes us as unsatisfactory.”161 Additionally,
Metaphors We Live By and More Than Cool Reason have limited bibliographies with less
than 20 entries apiece that conspicuously fail to include important works on metaphor that
are being argued against. Max Black notes, rightfully, that Metaphor We Live By’s “absence
of an index is deplorable.”162 These factors combine to make CMT scholarship — such
as the summary provided above — needlessly difficult and frustrating. Moreover, Lakoff
and his coauthors seem strangely reluctant to engage in exchanges with their peers, rarely
acknowledging or addressing even constructive criticisms.163 Lakoff’s scholastic style thus
diverges from the traditional modus operandi of the academy, where transparent, thorough
referencing and interactive debate and criticism among peers is the norm. In addition to
being put off by Lakoff’s unconventional scholarship, some readers may have been incensed
161
Ray Jackendoff and David Aaron, “Review of More Than Cool Reason,” Language 67.2 (1991): 321.
Max Black, “Review of Metaphors We Live By,” J. of Aesthetics and Art Criticism 40.2 (1981): 210.
It seems that Lakoff may have taken these criticisms to heart: Philosophy in the Flesh has 18 pages of
categorized references and a 22-page index, and Where Mathematics Comes From has 20 pages of each.
Unfortunately, in-text citations remain scant in more recent works.
163
I am aware of only two instances where Lakoff has provided an explicit defense against one of his
critics. One is his well-known response to Steven Pinker’s book review of Whose Freedom?: The Battle over
America’s Most Important Idea. The other is a paper coauthored with Mark Johnson responding to Martina
Rakova’s criticisms of their philosophy. Lakoff’s response is that many of the criticisms leveled depend on
assumptions that are called into question by CMT’s empirical research and that the critic fails to address
this incompatibility (Johnson and Lakoff, “Why cognitive linguistics requires embodied realism” 260).
162
97
by the sometimes inflammatory and partisan political theorizing that has become one of his
main foci over the last decade. While none of these criticisms are fatal to Lakoff’s theory,
they do make a contribution to the general apprehensiveness towards CMT found in much
of the literature.
As I mentioned at the beginning of the chapter, many of the philosophical criticisms
of CMT cluster around
CONCEPT
and fall into two categories. The first category involves
the charge that conceptual metaphor entails an unsatisfactory understanding of concepts.
Included in this category are objections that agree metaphor is a matter of thought, but hold
that Lakovian conceptual metaphor does not sufficiently explain the phenomena in question.
To defend against these criticisms, I will evaluate CMT using Prinz’s seven desiderata for
theories of concepts, and suggest ways in which its weaknesses may be overcome. I am not
particularly concerned with the subcategory of criticisms that reject the details of conceptual
metaphor because they concur that thought is metaphoric, which is the result I primarily
wish to defend. The second category of criticisms focuses on the connexion between concepts
and language in CMT. As Prinz intentionally excluded language desiderata from his list, we
shall have to look elsewhere for support. The work of Cornelia Müller takes an approach
that could help overcome these concerns.
In CMT, “concepts are neural structures that allow us to mentally characterize our categories and reason about them.”164 Categories are not mental representations of natural
kinds gleaned from the world, “sets defined by common properties of objects.”165 Rather,
Lakoff says that “human categorization is essentially a matter of both human experience and
imagination — of perception, motor activity, and culture on the one hand, and of metaphor,
metonymy, and mental imagery on the other.”166 His embodied-realist theory of conceptual
categories holds that thought, experience and, therefore, categorization essentially involve an
interaction between an organism and its environment.167 Though Lakoff extensively develops
this theory of categorization in Women, Fire, and Dangerous Things, delving into it would
be a lengthy, distracting tangent. Therefore, for the sake of simplicity and generality, I will
164
Lakoff and Johnson, Philosophy in the Flesh 19; emphasis theirs.
Lakoff, Women, Fire, and Dangerous Things 10.
166
Women, Fire, and Dangerous Things 8.
167
Johnson and Lakoff, “Why cognitive linguistics requires embodied realism” 249.
165
98
restrict my discussion as much as possible to aspects of the theory immediately connected
with conceptual metaphor as discussed above.
As a theory of concepts, CMT has several strengths. CMT does an excellent job of
satisfying the scope desideratum. The variety of cognitive structures and mappings it includes
— including, but not limited to, conceptual metaphor, conceptual metonymy, and imageschemata — can account for a wide range of conceptual kinds and levels of abstraction.
Should a concept outside of CMT’s scope be found, CMT is well positioned to rectify its
deficiency by incorporating new cognitive mechanisms; conceptual metaphor plays well with
a wide variety of conceptual mechanisms and structures. In Women, Fire, and Dangerous
Things, Lakoff puts forward a theory of concepts as idealized cognitive models, and several
chapters are devoted to discussing and accounting for a variety of different models — including
cluster models, radial models, and classical definitional models.168 The particulars of Lakoff’s
theory of categorization aside, any account of concepts that can accommodate conceptual
metaphor will include the idea that concepts have differing kinds and amounts of structure,
and will therefore be better able to satisfy the scope desideratum than the classical theory
of concepts.
CMT also does well with the categorization desideratum. Lakoff frequently emphasizes
that a core value of his project is having his theories account for or be consistent with
the body of experimental evidence.169 This commitment to being empirically responsible
is very apparent in Women, Fire, and Dangerous Things, which “surveys a wide variety
of rigorous empirical studies of the nature of human categorization” and integrates these
results into CMT.170 In particular, Lakoff emphasizes the importance of Rosch’s experimental
findings on prototype effects and basic-level categories in ushering in a new paradigm in
concept scholarship. Lakoff holds that conceptual metonymy can explain Rosch’s results: a
metonymy is invoked when a person thinks or reasons about a category in general by means
of a particular prototypical representative, for example.171 Despite Lakoff’s emphasis on
Rosch’s typicality evidence in the development of his theory, CMT is not an instance of a
168
Women, Fire, and Dangerous Things 153–4.
For example, see Lakoff and Johnson, Metaphors We Live By 246.
170
Lakoff, Women, Fire, and Dangerous Things 11.
171
Women, Fire, and Dangerous Things 79.
169
99
prototype theory as it rejects the idea that categories are defined by prototypes.172 Note
that avowing empirical responsibility does not necessarily mean that one’s theory actually
accounts for all the relevant facts; critics could object that CMT does not adequately account
for the empirical facts it claims to, or that it fails to take some particular piece of relevant
evidence into account.173 In general, such empirical criticisms seem more likely to suggest
directions for further development of the theory than to be devastating. Addressing questions
of the empirical adequacy of CMT is beyond the scope of my dissertation; a robust treatment
of the relevant scientific results would require a much-larger, book-length project.174
The summary of conceptual metaphor presented above suggests that CMT satisfies the
acquisition and publicity desiderata: concepts emerge from recurrent patterns in our experiences, and are public because those experiences involve our being similarly embodied in
a shared environment. One of the key strengths of CMT is that its emphasis on cognitive
mappings naturally provides an account of conceptual emergence and development, an aspect
that is often neglected by other philosophical theories. The account of acquisition provided
by Lakoff seems quite plausible, if complex and somewhat lacking in detail. I will therefore
devote my attention to claims that the publicity desideratum may not be successfully met.
Prinz argues that theory-theory fails to satisfy publicity because “[i]t is very unlikely that
any two people have exactly the same theories of the categories they represent. . . Theories
mushroom out to include all our beliefs about a category. Such large belief sets inevitably
differ from person to person.”175 One may argue that Lakovian idealized cognitive models are
enough like theories to fall victim to this criticism as well. However, the constraints imposed
by embodiment mean that idealized cognitive models cannot be as problematically ad hoc
and idiosyncratic as Prinz claims theories are.176 Moreover, while successful communication
requires that we possess shared concepts to some degree, this does not entail that everyone’s
concepts must be identical. Prinz considers this response — that conceptual similarity is sufficient for publicity so strict identity is unnecessary — and concludes that it is not in the spirit
172
Women, Fire, and Dangerous Things 136–7.
See McGlone (2007) for an example of a criticism of the second type.
174
See Gibbs (2011) for a recent defense of CMT from the perspective of a cognitive scientist.
175
Prinz 87.
176
Prinz 87.
173
100
of theory-theory. However, such an approach does seem to be consistent with, and therefore
available to, both prototype theory and CMT.177 Concepts differ between cultures as well as
between individuals and can cause communicative difficulty or even outright communicative
failure. One place that the effects of conceptual disagreement can be observed is in discussions about abstract notions, such as the afterlife. All of this suggests that Prinz’s original
formulation of the publicity desideratum may be too strong, a conclusion he endorses insofar
as the defense of his proxytype theory of concepts utilizes a relaxed version of the publicity
requirement that does not require strict identity between concepts.178 Thus, like proxytype
theory, CMT satisfies the publicity desideratum better than theory-theory insofar as it allows
for variation among individual’s concepts but commonalities of embodied experience provide
an intersubjective core to many concepts that allows for successful communication.
Michiel Leezenberg explicitly criticizes Lakoff for failing to account for conceptual publicity: he claims if conceptual metaphors precede their linguistic instantiations as CMT claims
then it is difficult to understand how “private concepts or experiences warrant that people
have the same public meanings.”179 In particular, Leezenberg is worried that CMT contains
a vicious circularity: pre-conceptual structure grounds conceptual structure, which allows
for language, which is necessary for culture, which plays an essential role in pre-conceptual
structure.180 While Leezenberg’s observation that Lakoff could spend more time considering
the details of how cultural factors contribute to concepts is well taken, his argument that
CMT fails to meet the publicity requirement seems problematic. He erroneously assumes
that concepts must be wholly structured prior to language use.181 The very idea that a
177
Prinz 88–9.
Prinz 158. After using his desiderata to establish the faults of various popular theories of concepts, Prinz
puts forward a theory of concepts as proxytypes, perceptually derived long-term memory networks that are
currently, or have the potential to be, activated in working memory as a mental representation of a category
(148–9). Unlike prototype theory and theory-theory, which identify conceptual structure as a single kind
of information, proxytype theory insists that multiple kinds of knowledge are involved in concept formation
(Prinz 314). Prinz also emphasizes that the representations necessarily retain their perceptual modality
(119). Thus we see that there are parallels between proxytypes and embodied idealized cognitive models.
Though Prinz and Lakoff do not mention these parallels, Lawrence Barsalou (whose work underpins proxytype
theory) explicitly acknowledges similarities between elements of his theory and Lakovian idealized cognitive
models (Barsalou 17). Reconciling Prinz and Lakoff on concepts could be fruitful, but the constraints of this
dissertation necessitate that such an undertaking happen elsewhere.
179
Leezenberg 141; emphasis his.
180
Leezenberg 142. A similar criticism occurs in Rakova (2002).
181
Leezenberg 142.
178
101
concept can be completely structured is problematic, as concepts are dynamic and have the
potential for adaptive development prompted by new experiences: humans learn! The dynamic interplay between conceptual metaphors, their experiential bases, and their cultural
and linguistic realizations that was discussed above suggests that the circularity that Leezenberg refers to is virtuous rather than vicious.182 Recall the discussion about circles (and
other regular polygons): geometric regularities in the environment are experienced, those
experiences lead to a primitive conceptualization of
CIRCLE,
that conceptualization allows
people to create representations of circles and to talk about them, and thus more circles and
circle-talk are introduced into the environment, the experience of which allows for further
development of the CIRCLE concept, and so on. Such virtuous conceptual circles exist at both
the individual and population levels, and they interact: pre-conceptual infants are born into
an environment pervaded with both linguistic and non-linguistic instantiations of concepts of
the humans who have come before them. In this sense, language can precede concepts. For
many concepts, a cultural contribution occurs only after a substantial conceptual core has
developed based on common biological experience. That is, while culture may indeed play
an important role in pre-conceptual structure as Leezenberg claims, it is clear that organisms
possess a variety of discriminatory mechanisms that also provide significant pre-conceptual
structure but are not culture dependent. Leezenberg’s philosophical arguments fail to take
into account the body of evidence Lakoff appeals to in suggesting that commonalities in
the embodied experiences of infants provides the ground for the public intersubjectivity of
concepts. Moreover, his concern that “the mere fact that two individuals live in the same
environment, and behave in the same way, gives no conclusive evidence as to their inner
workings”183 seems overly skeptical; while this observation has some merit, for our purposes
it can be adequately addressed by the theoretical virtues of simplicity and parsimony. Thus,
182
It may be fruitful to conceive of virtuous circularity as a helix rather than a circle, progressing upward
as it goes around rather than problematically returning to its origins. Another related way of conceiving
a virtuous circle that may be useful relies on the distinction between displacement and distance. If one
circumnavigates the Earth and ends at the same place they started, then their displacement is zero: rather
than traveling from point A to point B, they have traveled from point A to point A and haven’t “gotten
anywhere.” However, even if one ends up where they started, the distance covered going around the circle
can be extensive. Just as repeatedly traversing a circular trail can etch a path into the grass, going around
conceptual circles need not be an exercise in futility but rather can lead to conceptual development and
refinement.
183
Leezenberg 141.
102
while the details of CMT’s complex story of concept acquisition and grounding could use
further elaboration, it seems plausible that it satisfies the publicity desideratum given the
current state of the theory.
Whether CMT satisfies the compositionality criterion is debatable, largely because the
desideratum itself is somewhat contentious. Following Fodor, Prinz states that “[c]oncepts
are compositional just in case compound concepts (and thoughts) are formed as a function of
their constituent concepts together with rules of combination.”184 However, as was the case
with publicity, Prinz claims there are reasons to believe that this formulation of the compositionality requirement is too stringent. In particular, he says that the desideratum should be
interpreted as saying that a theory must allow for conceptual composition but that it need
not require that the entire content of a concept must be inherited from its constituents.185
When relevant background knowledge or experience with exemplars of a complex concept are
available, we should expect that such knowledge will often trump pure compositional rules
in providing structure.186 For example, most people’s understanding of GREEN APPLE includes
the notion
TARTNESS,
a notion that belongs to neither
APPLES
nor GREEN but is based on their
experiences of eating green apples. Prinz concludes that, contra Fodor, both prototypes and
proxytypes are compositional in this relaxed sense. Lakoff claims that “[w]ithin a theory that
contains basic-level concepts and image schemas, it is still possible to have rules of semantic
composition that form more complex concepts from less complex ones.”187 The sketch of
CMT provided above illustrates that one aim of the theory is economically explaining systematic conceptual structure by means of various cognitive mechanisms, including mappings
like conceptual metaphor and conceptual blends. Though Fodor would certainly disagree,
there are good reasons to think that CMT satisfies the relaxed compositionality desideratum.
Only the content desiderata remain. CMT seems well positioned to satisfy the cognitivecontent requirement (recall that this desideratum requires a theory of concepts to account
for the psychological and referential aspects of concepts that can not be explained in terms
of intentional content alone). The nature of embodiment means that concepts arise from our
184
Prinz 12.
Prinz 291.
186
Prinz 292.
187
Women, Fire, and Dangerous Things 280.
185
103
experiences, not as direct representations of the world. Experiences of the same object or
event can vary dramatically from person to person, and thus different concepts may refer
to the same thing; thus, canines may be “lovely puppies” for one person but “hell hounds”
for another. Indeed, the kind of “seeing-as” that occurs in conceptual metaphor — when
one conceptualizes love as a journey or a force or a work of art, for example — lends itself
to explaining the observed differences in our conceptualizations that motivate the cognitive
content desideratum. Explaining how CMT satisfies the desideratum of intentional content is
more difficult, as is true for many non-traditional theories of concepts. Prinz argues that the
intentional contents of many concepts are natural kinds, collections with boundaries independently determined by nature.188 His explanation of how concepts attain their intentional
contents involves a combination of nomological covariance (concepts involve detection mechanisms, which may be fooled by counterfeits) and etiology (incipient causes and histories of
concepts are a contributing factor in reference, helping eliminate problems coming from false
positives).189 It seems that CMT will not be able to co-opt Prinz’s arguments verbatim, as
Lakoff has repeatedly denied the existence of natural kinds.190 The relevant question then
becomes, what are the intentional contents of concepts according to CMT?191
Lakoff’s anti-objectivism and its associated rejection of natural kinds does not constitute
a wholesale abandonment of realism, as some may be inclined to believe. CMT and the
associated philosophical position known as embodied realism represent an attempt to find an
empirically responsible middle ground between the unpalatable extremes of objectivism and
radical relativism.192 CMT is not an objectivist position because it holds that it is impossible to eliminate the contribution of our embodied perspective from our understanding. Our
categories are part of our experience, not part of a human-independent objective reality.193
188
Prinz 4.
See Prinz (2002), chapter 9.
190
See, for example, Women, Fire, and Dangerous Things page 9, and Philosophy in the Flesh page 101.
191
Determining the nature of the connexion between thoughts or concepts and the world is a perennial
problem in philosophy; obviously it is not possible to definitively resolve it here.
192
Lakoff uses “objectivism” somewhat idiosyncratically to refer to a kind of unsophisticated realist thinking:
there exists an objective reality, our perceptions of that reality are direct but flawed reflections of it, and if we
could but overcome those flaws (through precise language and rigorous scientific methodology, for example),
we could possess absolute, unmediated, and definitive knowledge of the objective world (Metaphors We Live
By 186–7). Throughout this dissertation, “objectivism” should be interpreted in this way (and not be taken
to refer to Randian philosophy, for example).
193
Lakoff and Johnson, Philosophy in the Flesh 19.
189
104
On the other hand, CMT also rejects radical “anything goes” relativism. Humans have some
freedom in their conceptualizing, but “there are also a great many conceptual universals.”194
Thus, in CMT, categories are not natural kinds insofar as they are not mental copies of
clusters of properties inherent in the world, yet they are constrained, structured, and granted
stability to varying degrees by a world that is independent of humans via embodied interactions with that world.195 Though our categories are models in us rather than inherent
collections in the world, many of them arise unconsciously and automatically in all human
beings — albeit, by a more complicated mechanism than direct representational projection
— and could therefore be called natural. Moreover, it is possible to make categorizational
mistakes when dealing with these natural categories: models may be misapplied! Naturally
arising intersubjective consensus seems to be a suitable substitute for full-fledged, objective
natural kinds.196 The unconscious ease with which basic-level categories arise early on in our
conceptual development may be responsible for the idea they are objective natural kinds.197
The following passage from Lakoff and Johnson nicely summarizes their position on natural
kinds: “One can believe that objectivist models can have a function — even an important
function — in the human sciences without adopting the objectivist premise that there is an
objectivist model that completely and accurately fits the world as it really is.”198
It thus seems that though CMT denies the existence of natural kinds it posits a plausibly
suitable equivalent in the form of basic-level concepts. However, this is clearly not the entire
story. Many of our concepts do not refer to natural kinds insofar as the things they refer to
do not exist in the world:
GRIFFONS,
for example. In chapter 4, we will see that some people
194
Johnson and Lakoff, “Why cognitive linguistics requires embodied realism” 252.
Lakoff and Johnson, Philosophy in the Flesh 90. It is worth noting that a more sophisticated understanding of natural kinds that deemphasizes independence from human cognition may be compatible with
CMT. To the extent this is possible, parts of Prinz’s arguments may be used to defend CMT. However, it is
unclear whether such an understanding of natural kinds would be acceptable to Prinz, who claims to endorse
“a strong form of realism,” with natural-kind concepts referring to naturally delineated categories (Prinz 5).
196
Above, it was suggested that Prinz might be an objectivist about natural kinds. Elsewhere, however,
Prinz suggests that concepts actually have two kinds of intentional content. The real content of a concept
is an independent thing in the world and the nominal content is how we take a thing to be (Prinz 277).
It seems that this picture could be made compatible with CMT with a little tweaking. In particular, this
would involve deemphasizing the idea of “real essences,” though it seems Prinz would not be happy about
this (Prinz 282).
197
Lakoff, Women, Fire, and Dangerous Things 31–8.
198
Metaphors We Live By 219.
195
105
believe that mathematical concepts fall in this class. While traditional theories positing concepts to be reflections of structures inherent in the world may have difficulties handling such
vacuous concepts, CMT has no such difficulty.199 By understanding the intentional content
of concepts to be world-grounded interactive experiential models rather than worldly things
in themselves, CMT causes controversy by defying tradition but simultaneously situates itself
in a good position for explaining vacuous concepts and other puzzle cases.
CMT stands up quite well as a theory of concepts insofar as it is seems able to satisfy
Prinz’s desiderata. The above brief discussion shows that it excels at satisfying scope and
acquisition, while further details may be desired in the case of intentional content and the
related desiderata of publicity and categorization. Such details may already exist somewhere
in the expansive Lakovian corpus or they may need to be newly created; the above discussion
has shown that, if the latter is the case, a promising tactic may be to adapt Prinz’s approach
so that it fits with CMT. Gibbs’ claim that concepts should be viewed “not as fixed, static
structures but as temporary representations that are dynamic and context-dependent”200 also
recommends the creation of a Lakoff-Prinz hybrid theory. Showing that CMT satisfies — or
at least has the potential to satisfy — Prinz’s desiderata provides an initial defense against a
range of objections and criticisms. This positive approach is preferable to considering specific
criticisms of the Lakovian theory of concepts because of its efficiency. Indeed, many of the
criticisms in the literature involve misinterpretations of CMT that are best defended against
by removing the ambiguity or lack of understanding that facilitates them. One example of
such a criticism is Marina Rakova’s claim that Lakoff’s embodied realism is problematically
empiricist about concept acquisition, rejecting all traces of conceptual nativism; this would
make it difficult — if not impossible — for embodied realism to account for a variety of empirical results.201 Explaining how CMT satisfies Prinz’s desiderata shows that this criticism
is misguided insofar as it overlooks the central claim that all experiences involve interactions between the environment and embodied perceptual and cognitive capacities that are
199
Prinz brings this up as an important objection, especially against etiological theories of intentional content
(239).
200
Raymond W. Gibbs, “Why many concepts are metaphorical,” Cognition 61 (1996): 313.
201
Martina Rakova, “The philosophy of embodied realism: A high price to pay?” Cognitive Linguistics 13.3
(2002): 219.
106
not derived from experience; embodied realism is demonstrably not extremely empiricist and
antinativist as Rakova claims.202 I will conclude this chapter by considering a few specific
objections against Lakoff and make some suggestions about how they may be overcome.
One objection that has been raised by several authors is that Lakoff often argues against
straw men. Leezenberg presents this as a continuation of the criticism that Lakoff rarely
argues against specific authors, claiming that “[t]he ‘objectivist tradition’ they fulminate
against is not ‘fundamentally misguided’ or ‘humanly irrelevant’ but simply nonexistent.”203
At times, Lakoff’s antiobjectivist arguments exhibit a seemingly hyperbolic character. However, Lakoff is not always so grandiose:
the myth of objectivism is not itself objectively true. But this does not make
it something to be scorned or ridiculed. The myth of objectivism is part of the
everyday functioning of every member of this culture. It needs to be examined and
understood. We also think it needs to be supplemented — not by its opposite, the
myth of subjectivism, but by a new experientialist myth, which we think better
fits the realities of our experience.”204
Elsewhere, Lakoff explicitly acknowledges that his arguments are aimed at a general, idealized
paradigm rather than the nuanced views of specific individuals.205 While Leezenberg is
almost certainly correct that no contemporary philosophers adopt a hard line, capital-T
Truth objectivist stance, Lakoff also seems to be correct in claiming that various objectivist
doctrines are still pervasive, both in our everyday thinking and in our academic pursuits. It
is important to note that the straw man objection is one of several criticisms focusing on
Lakoff’s arguments against objectivism, but, apart from playing a motivational role, these
arguments are independent from the constructive theorizing behind CMT and embodied
realism.206 Therefore, whether one believes that the straw man objection is successful or not
has little bearing on the positive claims of CMT which are the subject at hand.
Another criticism leveled against CMT is that many of the key notions involved in the the202
Johnson and Lakoff, “Why cognitive linguistics requires embodied realism” 247–51.
Leezenberg 137. Murphy expresses a similar view, though perhaps somewhat more charitably than
Leezenberg: “This monolithic view [that Lakoff and Johnson call objectivism] is one that many psychologists
would not want to commit themselves to” (“On metaphoric representation” 179).
204
Lakoff and Johnson, Metaphors We Live By 186.
205
Women, Fire, and Dangerous Things 157.
206
Leezenberg 137.
203
107
ory (culture, meaning, and imagination, for example) are carelessly defined.207 If one accepts
the clichéd aphorism “the first rule of philosophy is ‘define your terms’ ” this alleged carelessness would seem to be an egregious oversight. In making this criticism, Leezenberg seems
to erroneously assume that CMT is exclusively philosophical rather than an interdisciplinary
theory based upon empirical evidence. While this observation may weaken the criticism, it
does not altogether defeat it. However, there is reason to believe that this is objection is
not devastating. The above tenet is strongly tied to the classical theory of concepts that is
incompatible with CMT’s observation that much conceptual structure is non-definitional. In
particular, the terms singled out by Leezenberg are examples of abstract concepts with structure that defies straightforward definition. Despite attracting a certain amount of criticism,
the fact that CMT not only allows but sometimes requires a certain looseness in interpretation of terms seems to be one of the strengths of the theory, as it allows for a certain amount
of compatibility with what might appear to be competing theories.208 Correspondingly, as
the previous paragraph notes, some of the most objectionable passages in the CMT corpus
occur when Lakoff makes overly definitive claims that conflict with this flexibility. However, many pages have been written since CMT’s inception with the aim of gently clarifying
key notions without becoming contradictorily exact; this work is often a frustratingly slow
and lengthy process, but has been relatively successful in providing a clearer understanding
than was available when it started. While further careful explanation of the core concepts
in CMT is certainly desirable, careless attempts to provide exact definitions of the terms
involved might contradict the Lakovian theory of concepts.
A range of criticisms focus on specific conceptual metaphors discussed in the Lakovian
corpus. One such criticism holds that certain conceptual metaphors are worryingly arbitrary,
since alternative, related conceptual metaphors exist that could perform the same explanatory
role equally well. For example, Jackendoff and Aaron question why
have priority over
LIFE IS SOMETHING THAT GIVES OFF HEAT
or
LIFE IS A FIRE
LIFE IS A FLAME.
209
would likely respond that the priority in this case derives from the fact that
should
CMT theorists
FIRE
is a basic-
level concept, whereas the other two source domains are more general and more specific
207
See, for example, Leezenberg, page 138 and Black, “Review” page 209.
This property of CMT is put to good use in chapter 4.
209
Jackendoff and Aaron 324.
208
108
respectively. Additionally, Lakoff and Johnson stress that the notations they use to refer
to conceptual metaphors are names for, rather than accurate descriptions of, the underlying
mappings.210 Thus, it is possible that the three different metaphor names above refer to the
same underlying mapping. Another type of criticism claims that some specific attribution
of linguistic utterances as symptoms of an underlying conceptual metaphor is post-hoc and
erroneous, either because a different conceptual metaphor is actually responsible or because
the mapping named does not actually exist or is not metaphorical in nature.211 Max Black,
for example, claims that
TIME IS MONEY
is not a metaphor but rather the sentence “Time is
money” is merely an adage.212 . Finding some conceptual metaphor to be implausible or even
outright incorrect does not constitute an objection to CMT but only one piece of potential
evidence counting against it. Even if the details of all of the specific conceptual metaphors
presented in the Lakovian corpus turn out to be incorrect, this does not constitute a definitive
refutation of CMT; that is, a failure to correctly describe any existing conceptual metaphors
does not rule out the possibility that such mappings do actually exist. Moreover, while
the names of Lakoff’s metaphorical mappings can make them seem spurious or optimistically
simplistic, some of them have been the subject of extensive research that has provided a body
of evidence supporting their existence; examples include Michael Reddy’s
CONDUIT
metaphor
and Lena Boroditsky’s work on time metaphors.213 Thus, while these objections to specific
conceptual metaphors should be considered and addressed by CMT researchers, they are not
fatal to the theory.
A related objection, first noted by Black, contends that conceptual metaphor analysis
contains problematic circularity.214 Murphy states the problem in the following way: “linguistic data are used to identify metaphors, but the main concrete predictions the theory
makes are about similar linguistic and psycholinguistic data.”215 McGlone expresses the same
concern, advising conceptual metaphor theorists to “abandon circular reasoning. . . and seek
210
Philosophy in the Flesh 58.
Jackendoff and Aaron 330–1.
212
“Review” 210
213
See Reddy (1979) and Boroditsky (2000) for more details.
214
Black, “Review” 209.
215
Murphy, “On metaphoric representation” 200.
211
109
substantiation of their claims that is independent from the linguistic evidence.”216 Regardless of how plausible or elegant a theoretical posit may be, empirical legitimacy requires that
it make testable predictions about phenomena sufficiently distinct from those that motivate
it. Murphy suggests that CMT can be strengthened in this respect by providing “predictions about memory, problem solving, induction, measures of conceptual structure (such as
typicality and categorization), learning and performance.”217 While CMT may have been
vulnerable to this objection in its origins, more recent work has worked towards overcoming this objection. Lakoff and Johnson note that there are at least nine types of empirical
evidence in support of CMT and, while most of these are linguistic in nature, they involve
notably different facets of language.218 One of the nine types, spontaneous gesture study,
is of particular interest because it provides a nonlinguistic source of evidence; experiments
show that the unconscious gestures people make while speaking are frequently consistent
with the conceptual metaphors posited to be behind the utterances synchronous with the
gestures. For example, while talking about balancing a variety of viewpoints, one subject
was observed to hold their hands out with palms cupped upwards, and move them alternately
up and down, gesturally mimicking a set of scales; in this case, both the utterance and the
gesture are interpreted as realizations of the conceptual metaphor
IMPORTANCE IS WEIGHT.
219
This observed multimodality supports the idea that Lakovian metaphors reach deeper than
language and structure our concepts. Gallese and Lakoff’s 2005 paper makes some important
initial steps towards grounding CMT in neurobiological evidence, and includes a discussion
of some future experiments that could help empirically validate their theory.220 Though the
nonlinguistic evidence in favour of CMT may be too scant at present for some critics, recent
developments suggest that the situation may improve in coming years.
Another objection to CMT is that it “enlarges the scope of the term ‘metaphor’ well beyond the standard use of the term.”221 While a cursory perusal of Lakoff’s work may give the
216
McGlone 115.
Murphy, “On metaphoric representation” 200.
218
“Why cognitive linguistics requires embodied realism” 250.
219
Müller 96.
220
Vittorio Gallese and George Lakoff, “The Brain’s Concepts,” Cognitive Neuropsychology 22.3 (2005):
471.
221
Jackendoff and Aaron 325.
217
110
impression that all thought is metaphorical, a more thorough and careful reading shows this
to be false. Metaphor is extremely pervasive according to CMT but it stops short of being
ubiquitous: the term is not vacuous but picks out a useful distinction. Concepts have varying
amounts of implicit, non-metaphorical structure, and can be said to be “literal” to the extent
they are comprehended or utilized without any use of conceptual metaphor. Some primitive concepts are entirely non-metaphorical, while even highly abstract concepts like
LOVE
have a scant literal skeleton.222 Sentences are non-metaphorical when their interpretation
requires only literal facets of concepts; Lakoff says the sentence “the cat is on the mat” is not
metaphorical.223 However, a new problem arises: how does one decide whether a particular
instance of language should be interpreted as metaphorical? Leezenberg states the problem in
the following way: “[Lakoff and Johnson] reject the ‘objectivist’ accounts that treat metaphor
as semantically deviant or arising from a defective ‘literal meaning,’ but offer no clear alternative, even though it is obviously necessary. After all, some sentences can be interpreted
either literally or metaphorically, and the same sentence may receive different metaphorical interpretations in different contexts.”224 CMT has several avenues of response available.
First, recall that CMT claims that the majority of conceptual metaphor use is automatic
and unconscious; therefore, the production and interpretation of metaphorical sentences in
CMT do not require conscious recognition of metaphoricity the way other theories do and,
to the extent this is true, the objection simply does not apply to CMT. However, given that
we do consciously recognize sentences as literal or metaphorical on occasion, and given that
the interpretation of metaphorical language sometimes requires conscious effort, something
should be said about linguistic metaphor detection in CMT. Contrary to Leezenberg’s claim,
Lakoff comments that there are occasionally cases where a metaphorical interpretation is
algorithmically derived from the literal meaning; unfortunately, he does not elaborate and
we are thus forced to speculate what he might have meant by this.225 One mechanism by
222
While Lakoff tells us “there is. . . an extensive range of non-metaphorical concepts” (“Contemporary
Theory of Metaphor” 205), he provides no specific examples. The most basic concepts, such as UP, seem
to be the likely candidates, but even such concepts with significant amounts of direct experiential structure
have the potential to receive additional metaphorical structure.
223
Lakoff, “Contemporary Theory of Metaphor” 205.
224
Leezenberg 139. Similar charges are levied by Mac Cormac (1985), Jackendoff and Aaron (1991), and
McGlone (2007).
225
“Contemporary Theory of Metaphor” 205.
111
which an audience may come to interpret a sentence metaphorically is if the author of the
sentence explicitly directs them to do so (“She has a monkey on her back, metaphorically
speaking”). An author may also direct their audience to a metaphorical interpretation implicitly through phonological or graphological emphasis, associated gestures, or particular
sentence construction (including, but not limited to, word choice and word order).226 For
example, the combination of adjective choice and emphasis in the sentence “If my flighty
niece is late again, I shall have to clip the wings of the butterfly” suggests an interpretation
utilizing the metaphor MY NIECE IS A BUTTERFLY.227 Though Lakoff consistently rejects the traditional literal/metaphorical dichotomy, in an oft-overlooked paper he contends that many
of the problems with that distinction arise because the standard understanding of literality
conflates four distinct and non-convergent senses in which an instance of language can be
literal:
1. Conventional literality occurs when language is used in an unembellished, straightforward, and direct way.
2. Subject-matter literality involves talking about some domain of subject matter in
the ordinary way.
3. Nonmetaphorical literality occurs when an instance of language is directly meaningful and interpreted without the use of any conceptual metaphors.
4. Truth-conditional literality is exhibited by utterances which are capable of being
objectively true or false.228
Of these senses, only nonmetaphorical literality contrasts with linguistic metaphoricity.229
The opposition between nonmetaphorical literality and metaphoricity is precisely the distinction invoked at the beginning of this paragraph; it is possible that more details of the
conscious, contextual application of this distinction could be obtained. Also, though Lakoff
226
Support for this idea can be found in Lakoff’s work: “Grammar can also play a role in activating a
metaphor” (“The Neural Theory of Metaphor” 35). See the quotation in the next paragraph for more
specifics about how Lakoff currently thinks metaphorical language works.
227
This metaphorical sentence is modified from an example of Martinich (“A Theory for Metaphor” 448).
228
George Lakoff, “The Meanings of Literal,” Metaphor and Symbolic Activity 1.4 (1986): 292.
229
Lakoff, “The Meanings of Literal” 295.
112
argues that the four senses of literality are non-convergent, this does not entail that they are
mutually exclusive; if one or more of the other senses of literality fails to obtain for some
sentence, this may sometimes suggest that exploring metaphorical interpretations is appropriate. Thus, there are several promising avenues that CMT could follow in trying to answer
the interpretation question.
Many of the specific criticisms presented thus far are connected to the general issue of
how concepts and language are related. That these problems require individual attention is
partially a consequence of Prinz’s omission of language requirements from the list of desiderata used to evaluate CMT in this chapter. The sketch of CMT provided above is consistent
with what Gibbs calls the cognitive wager, a commitment to the assumption that linguistic
structures have their basis in more general conceptual and experiential features of cognition;
this commitment places the burden of proof with those who make the contrary assertion that
language is autonomous from other modes of thought.230 CMT portrays sentences as mere
symptoms of underlying cognitive structures and mechanisms; thus, there seems to be little
relevant difference between a Shakespearean sonnet and the semi-articulate ramblings of a
simpleton foreigner insofar as both pieces of language invoke the same conceptual metaphor,
for example. Lakoff’s more recent, neurology-oriented writings seem consistent with this
picture:
The neural theory of language allows us to understand better why language is so
powerful. Let’s start with words. Every word is defined via linking circuit to an
element of a frame — a semantic role. Because every frame is structured by a
gestalt circuit, the activation of that frame element results in the activation of the
entire frame. Now, the frame will most likely contain one or more image-schemas,
a scenario containing other frames, a presupposition containing other frames, may
fit into and activate a system of other frames, and each of these frames may be
structured by conceptual metaphors. All of those structures could be activated
simply by the activation of that one frame element that defines the meaning of
the given word. In addition, the lexical frame may be in the source domain of a
metaphor. In that case, the word could also activate that metaphor. In the right
context, all of these activated structures can result in inferences.231
This description of the connexion between words and neural concepts is relatively detailed
compared to other related passages from earlier in the CMT corpus; however, it is consistent
230
231
Raymond W. Gibbs, The Poetics of Mind (New York: Cambridge UP, 1994), 15.
Lakoff, “The Neural Theory of Metaphor” 34.
113
with the earlier theory in that its emphasis on the conceptual contributes to an underemphasis
on the specifics of the linguistic side of the relationship, as Cornelia Müller here describes:
Remember that the ultimate goal of Lakoff and Johnson’s enterprise is, at its
core, an experiential theory of human understanding that is based on processes
of figurative thinking. . . language is primarily of interest insofar as it provides
insights into the functioning of the mind and more specifically into general cognitive mechanisms. These are then projected back to explain verbal metaphorical
expressions in terms of conceptual metaphors. This procedure, important as it
is, necessarily ignores linguistic variation. Its focus is on the common principles
that underlie language and language use and not on variation, be it caused by
differences in interindividual usage or in the conditions under which language is
used. . . [t]he assumed conceptual metaphors may well be plausible on the level of
a language community and probably also on the level of a collective mind. Yet
the extent to which they guide the understanding and actions of individuals is a
fundamentally different question.232
Authors like Gibbs and Müller, though generally sympathetic to the Lakovian position,
criticize CMT for this lack of balance and its dearth of details in describing the relationship
between the conceptual and the linguistic, and use these shortcomings as motivation in their
own research. I will conclude this section on the problems facing CMT by briefly discussing
two promising avenues that may help overcome this final criticism.
In Metaphors Dead and Alive, Sleeping and Waking, Müller claims that “the distinction
between dead and live metaphors is indeed at the core of traditional as well as of cognitive
metaphor theories [but]. . . it has not received the profound attention of metaphor scholars,
resulting in the lack of its systematic integration into theories of metaphor.”233 One of the
foundational elements of her theory is a “distinction that metaphor theories tend not to reflect
upon systematically: the distinction between the collective level of linguistic and/or conceptual systems and the individual level of representation and actualization of those systems.”234
In speaking about metaphors at the “level of the system,” Müller is in part referring to words
and phrases as they might feature in a lexicon or a phrasebook, relatively fixed in meaning
or interpretation (at a given point in time) and detached from specific authors, audiences,
and situations. However, it seems that “system” must refer to something more than this
232
Müller 47.
Müller 10.
234
Müller 12–3.
233
114
given that she claims metaphors may also be instantiated in non-linguistic modalities such
as picture and gesture. Hence, Lakovian conceptual metaphors would seem to constitute
paradigm cases of metaphor at the level of a system: conceptual metaphors are modalityindependent, intersubjective (or at least potentially intersubjective), and “subject to general
cognitive principles.”235 Whether the system involved is considered to be strictly linguistic
or more broadly cognitive or conceptual, Müller claims that for most theories of metaphor,
the “relation between the level of the system and the level of individual use is conceived of as
relatively unproblematic,”236 with particular instances of metaphor in discourse oftentimes
being regarded as merely derivative or epiphenomenal. Due in part to this frequent bias
toward the systematic in the literature, Müller decides to focus her attention on the level of
use in her book; however, she is careful to note that “[t]his is not to say that the perspective
of individual use should replace the systems perspective or that with regard to metaphors it
should replace semantics with pragmatics. Rather, individual use must be considered as a
noteworthy dimension of language with its coherent structures and organizational principles
and not just as a defective instantiation of whatever system.”237 That is, by focusing her
attention on the level of use, Müller is attempting to bridge the gap between system and use,
not to suggest the systems level is eliminable or otherwise unworthy of consideration.238 She
describes her position as follows:
this view complements the concept of conceptual metaphor with a level of verbal
metaphor and complements the concept of metaphor as a uniquely verbal phenomenon with a level of metaphor that is subject to general cognitive principles.
Hence, it provides support for the general claims of conceptual metaphor theory,
while at the same time pointing out the necessity of postulating a linguistic level
of metaphoric structure.”239
Müller’s theory shows promise in several ways. It is a fundamentally dynamic view of
metaphor with a firm commitment to empirical sensitivity. It addresses a wide sample of
traditional and contemporary theories of metaphor with uncommon objectivity, pointing out
their deficiencies while attempting to systematically incorporate — or at least be compatible
235
Müller
Müller
237
Müller
238
Müller
239
Müller
236
16.
13.
14.
15.
16.
115
with — their various strengths rather than rejecting them wholesale. The resulting synthetic
amalgam incorporates multiple dimensions (system versus use, language versus thought) and
multiple realms (conceptual, verbal, verbo-gestural, verbo-pictorial) of metaphor, and yields
a satisfyingly non-reductive, complicated, and balanced picture of metaphor as possessing
multiple levels of organizational structure.240 Though it is neither feasible nor necessary to
discuss the details of her theory here, it warrants further consideration by those concerned
with developing a unified theory of language, concept, and metaphor. In particular, the idea
of forming a hybrid of CMT and Müller’s theory warrants further attention.
Another promising approach would involve attempting to integrate semiotic insights with
CMT. Semiotics is the theory of signs, and their interpretation and communicative use.
Because signification permeates human existence, semiotics is a loosely unified discipline
with broad application. One traditional understanding of semiotics comes from C.S. Peirce:
“[Semiosis is] an action, or influence, which is, or involves, an operation of three subjects,
such as a sign, its object, and its interpretant, this tri-relative influence not being in any
way resolvable into an action between pairs.”241 This definition shows that language (signs),
concepts (interpretants), and the connexion between them are clearly within the purview of
semiotics, though they do not exhaust its scope. In semiotic terms, a metaphor can be seen
as resulting when the sign of one act of semiosis becomes the object of a secondary semiosis.
While this understanding may seem overly simplistic at first, unpacking its implications is a
daunting task that is best performed by semioticians; Umberto Eco has done some important
work on this problem.242 I will not go into the details of this account here. There are several
reasons why I think that attempting to integrate semiotics and CMT could be productive.
CMT has been criticized for overemphasizing the conceptual; making it compatible with the
semiotic trinity could create a stronger, more balanced theory. On the other hand, unlike
purely linguistic theories of metaphor, any semiotic account must involve discussion of interpretants, and is therefore likely to agree with conceptual theories in at least some respects. It
is plausible that Müller’s work could be compatible with an explicitly semiotic approach and
that the two could work in concert to improve CMT. The introduction of semiotic elements
240
Müller 112.
Qtd. in Eco, Semiotics and the Philosophy of Language 1.
242
See Eco, Semiotics and the Philosophy of Language, chapter 3.
241
116
could also fruitfully rejuvenate CMT’s relationships with some of its theoretical ancestors.
The irreducible trinity of Peirce’s definition is often represented pictorially as the triangle of
reference, popularized by Ogden and Richards in The Meaning of Meaning.243 As Richards is
often given credit for Lakoff’s core precept that “metaphor is primarily a matter of thought,”
the establishment of further consistencies between their work would be a positive development. Going further into history, John Kirby claims that “a semiotic presentation of the
Aristotelian model can be made congenial to Lakoff’s cognitive approach.”244 From the perspective of this document, the most significant advantage of combining CMT and semiotics
would be that it would facilitate a productive interaction between Lakoff and Núñez’s embodied theory of mathematics and Brian Rotman’s semiotic theory of mathematics. Though
I am optimistic that the suggestion of reconciling CMT and semiotics could help overcome
many of the problems facing the Lakovian theory, I am aware that such an undertaking would
take significant effort and patience, and may even be outright impossible. Mercifully, there
is no need for me to attempt this; an airtight defense of CMT is not required here.
In chapter 2, a historical sample of purely linguistic theories of metaphor was analyzed
and found wanting in various ways. In hopes of bypassing these problematic shortcomings,
chapter 3 considered the idea that metaphor could be conceptual in nature. Rather than
survey a variety of positions as before, I focused my attention on the foremost theory of
conceptual metaphor: Lakoff’s CMT. A synopsis of this theory distilled from the voluminous
corpus explained how the stable commonalities of human biological embodiment and the
environment we occupy combine with the capacity for creating mental metaphorical mappings
to yield abstract concepts, successful communication grounded in an intersubjective language,
and other facets of human experience. Prinz’s desiderata show CMT to be a promising theory
of concepts; while it has a few weaknesses, they seem non-fatal and no worse than those
observed in other leading theories of concepts. More-telling criticisms target the overemphasis
of CMT on concepts, and the corresponding scarcity of details regarding language use and the
relationship between the linguistic and the conceptual. In particular, some critics object that
CMT does not adequately explain under what circumstances a specific linguistic utterance
243
C.K. Ogden and I.A. Richards, The Meaning of Meaning (New York: Harcourt, Brace, and Co., 1923),
11.
244
Kirby 538.
117
receives a metaphorical interpretation. Observations such as this lend credence to claims
that CMT is not a sufficient theory of metaphor because not every example of metaphoricity
can be adequately construed as a symptom of conceptual metaphor.245 To overcome such
criticisms, I suggest that CMT might look to Prinz’s theory of concepts as proxytypes,
Müller’s theory of metaphor, or semiotics for inspiration. Indeed, many of the authors
criticizing CMT for not giving a complete account of metaphor concede that the insights
and evidence of the Lakovian position are at least somewhat meritorious, and incorporate
some notion of conceptual metaphor into their own theories.246 Thus, I have not provided an
airtight defense of CMT and I have conceded the possibility that better theories of metaphor
may exist. Has this entire chapter then been a pointless exercise?
As you undoubtedly surmise, the answer to this question is ‘no.’ While I believe that
pursuing a comprehensive philosophical account of metaphor is a worthy goal, it is not the
principal aim of this chapter. The primary goal of the first half of the dissertation was to
provide a summary of the philosophy of metaphor tailored to the task of examining possible relationships between metaphor and mathematics, to be undertaken in the second half.
Chapter 3 focused on assessing whether metaphor can be understood as conceptual, a matter
of thought rather than simply a matter of language. An affirmative result has been obtained:
while the stronger claim of CMT that every metaphor has a conceptual basis has not been adequately verified, there are good reasons to believe that at least some conceptual metaphors
exist. It is adequate for the purposes of the following chapters that the reader concede
that some conceptual metaphors may exist and constitutively structure abstract concepts,
including those of mathematics, a relatively weak requirement. Given the formal nature of
mathematical language, it seems likely that if metaphors play a role in mathematics, they
are more likely to be conceptual than linguistic in nature.247 I have focused discussion in this
chapter exclusively on CMT despite the existence of strong competing theories of conceptual
245
See, for example, Jackendoff and Aaron: “We are. . . not convinced that the notion of [conceptual
metaphor] characterizes a unified cognitive phenomenon. In fact, having drained from the term ‘metaphor’
much of its traditional content, [Lakoff and Turner] have created a theoretical construct so broad and unstructured that the term ‘metaphor’ may no longer be appropriate” (331).
246
For example, see Müller: “the fundamental claims of Lakoff and Johnson’s conceptual metaphor theory
now appear to be empirically validated” (51).
247
Or, if linguistic metaphors do play a role, they are likely to be very different from the poetic metaphors
put forward as paradigm cases by most theorists.
118
metaphor, including those of Müller, Leezenberg, Jackendoff, Gentner, and Fauconnier and
Turner. Note that nearly any theory that posits the existence of conceptual metaphors ought
to meet the weak requirement for this chapter mentioned above, so the exact details of how
conceptual metaphor is understood are not particularly important; thus, a survey of competing views is unnecessary.248 Given the lack of consensus in the field, I opted to focus on CMT
because of its various strengths: its extensive, systematic corpus; its empirical sensitivity;
its relatively clear presentation; its emphasis on human biology; and, significantly, the fact
that Lakoff has published extensive arguments that mathematics is founded on conceptual
metaphor.
248
If definitive evidence for one of the competing theories arose, I am confident that the position taken in
this dissertation could be modified to accommodate the new results.
119
Chapter 4
Mathematics is Metaphorical
La Mathématique est l’art de donner le même nom à des choses differéntes.
(Mathematics is the art of giving the same name to different things).1
— Henri Poincaré
In this chapter, I argue that mathematics constitutively involves metaphor, that it is
philosophically fruitful to conceive of mathematics as metaphorical in some important sense.
Though this idea is not a new one — various authors have espoused theories of mathematics
in this vein — it remains relatively unknown in both popular and academic circles and
is controversial among philosophers and mathematicians alike. This chapter defends the
Lakovian theory of embodied mathematics, in part by establishing coherences with other
theories of mathematics.
It is clear that the claim that mathematics involves metaphor depends heavily on how
these key terms are understood. The main purpose of chapters 2 and 3 was to paint a
picture of metaphor as a more complex, diverse, and legitimate phenomenon than the traditional view would have us believe. An acknowledgment that metaphors may sometimes
be more than eliminable obfuscating ornamentation is necessary if the above claim about
mathematics is not to be rejected as trivial and uninteresting, or outright preposterous. In
particular, cautious acceptance of the idea that some metaphors could be conceptual rather
than merely linguistic provides substance to the claim, insofar as mathematics is a highly
conceptual discipline with a frequent emphasis on precise formal language. Adopting a broad
and inclusionary understanding of metaphor as chapter 3 suggests is desirable as it allows
for an interesting range of possibilities.
1
Henri Poincaré, “The Future of Mathematics,” The Monist 20.1 (1910): 83.
120
What is mathematics? While this question may be easier to answer than the corresponding one about metaphor, it is not entirely straightforward and requires some explicit
attention. Most people know mathematics as a topic they are forced to study in school,
a subject consisting of facts and skills pertaining to numbers and shapes. Some of those
who do not find the discipline too onerous or repugnant will go on to become professional
mathematicians; others, such as scientists and engineers, will rely heavily on mathematical
techniques and concepts in their work. Mathematics is not confined to the classroom and
the laboratory, however, but is encountered and used by most people on a daily basis, in
their financial dealings, creative projects, and time management, for example. Two important ways of seeing mathematics emerge from this brief sketch. One is mathematics as a
subject matter, a collection of theories or a system of knowledge. The other important way
of seeing mathematics is as a collection of practices, such as proving and calculating; that is,
mathematics is something that one does.
The approach adopted in this chapter emphasizes mathematics-as-practice. Traditionally, the philosophy of mathematics has focused on mathematics-as-body-of-knowledge and
the associated ontological and epistemological issues. Such approaches often consider mathematical practice to be an unfortunate necessity, merely a clumsy means to an elegant and
austere end; this point of view has its roots in Plato. However, several perennial problems
plague traditional philosophies of mathematics, leading to a lack of consensus in the discipline and suggesting that a paradigmatically different approach is worthy of consideration.
Developments in the biological and social sciences over the last century have replaced a void
with an ever-increasing body of evidence about mathematical practices. These results help
motivate and shape practice-oriented approaches, insofar as it is desirable to have a theory
of mathematics that is compatible with the emerging data. Several practice-oriented theories of mathematics have arisen in recent years, though such alternative positions are still
very much a minority. Key examples include Lakoff and Núñez’s embodied mathematics,
Lakatos’s quasi-empiricism, Rotman’s semiotic approach, and certain strains of mathematical fictionalism. The goal of this chapter is not to provide a novel philosophy of mathematics,
but rather to show how the combination of practice orientedness with a progressive understanding of metaphor provides a framework that avoids some classic problems while allowing
121
for the reconciliation of the best features of multiple theories, both traditional and contemporary. Such an approach will have implications for the ontology and epistemology of
mathematics, though answering the traditional demands of these philosophical disciplines is
not a primary objective. The aims of this chapter are primarily descriptive, and do not aim
to alter mathematical practices that are successful. However, the frustration and loathing
that frequently accompany the learning of mathematics suggest that educational practices
may not be as successful as they could be; it will be suggested that embracing a practiceand metaphor-oriented approach could help alter teaching methods for the better.
The itinerary for this chapter is as follows. A brief exposition of the strengths and weaknesses of key traditional philosophical theories of mathematics provides background context
for the ensuing discussion. Lakoff and Núñez’s theory of embodied mathematics based upon
CMT is offered as a promising alternative that accounts for and integrates many of the positive insights of the traditional theories. The presentation of this viewpoint involves discussion
of some recent scientific findings concerning mathematical cognition, as such evidence is foundational to the theory. Like the underlying CMT, the theory of embodied mathematics is
controversial and has been criticized by both philosophers and mathematicians. It is shown
that many of these criticisms involve a fundamental misunderstanding deriving from the
fact that they cross paradigms. The chapter concludes with an examination of some other
contemporary theories of mathematics that involve metaphor and/or are practice-oriented.
In particular, Yablo’s mathematical figuralism, Rotman’s semiotic account of mathematics,
and Lakatos’s quasi-empiricism are discussed. Compatibilities between Lakovian embodied
mathematics and these other positions are indicated, suggesting directions for future research
and the development of a more robust metaphor-dependent account of mathematics.
4.1
Traditional Theories of Mathematics
The first section of this chapter is a short introduction to twentieth-century philosophy of
mathematics.2 The primary aim of this section is to provide a motivational context for alter2
Philosophizing about mathematics goes back at least as far as the Pythagoreans. While it would be ideal
to examine the entire history of the topic, such an approach is too extensive an undertaking for this chapter.
Fortunately, it will be adequate for most purposes of this chapter to view the philosophy of mathematics as
122
native practice-oriented approaches by exposing flaws in the leading traditional theories. The
goal is not to rigorously and exhaustively demonstrate that no adequate philosophical theory
of mathematics currently exists, nor to definitively reject all formulations of the traditional
approaches, but only to show that there is no consensus in this controversial discipline and
that there are serious difficulties that each of the classical theories must address if they are
to be considered tenable. There is a significant literature available to those desiring a more
detailed discussion of the controversy between the traditional positions than is given here.3
An additional motivation for considering alternative positions that comes out of this section
is that the hypothesis that metaphor plays a constitutive role in mathematics does not cohere
well with the traditional theories; this is far from a fatal objection but, for the purposes of
this chapter, does provide a reason to consider some less mainstream theories.
The philosophy of mathematics stands at the crossroads of several major areas of philosophy, including metaphysics, epistemology, logic, and the philosophy of language. It emerges
as a distinct philosophical subdiscipline because of the unique character of mathematics
among the various areas of human knowledge: it seems more rigorous, more abstract, more
universal, and more timeless.4 Accordingly, the central problems in the philosophy of mathematics consider how and why mathematics has this distinctive character, or, if it does not,
why it seems to. Traditionally, attempts to answer these problems involve discussion about
the nature of mathematical objects, the nature of mathematical truth, and the relationships
between these objects, truths and ourselves.5 Philosophical theories of mathematics can be
evaluated both on how well they address these key issues, as well as the extent to which they
cohere with our general understanding of non-mathematical objects and truth. This general
understanding provides a framework for evaluating and comparing theories of mathematics.
Though mathematical platonism has its roots in Plato’s theory of Forms, the former has
originating with Frege. However, the reader (and the author) should remain cognizant that, in actuality, the
tradition extends much farther into the past and Frege did not invent the subject ex nihilo.
3
For an exemplary single-volume treatment of these issues, see The Oxford Handbook of Philosophy of
Mathematics and Logic (2005).
4
It should also be noted that there is an important historical dimension to this: mathematics was recognized as a distinct area of knowledge relatively early on, and has had a place serving as the preeminent model
for knowledge since the origins of philosophy. Indeed, the ancient Greek root of the word “mathematics”
is µὰθηµα, which is translated as “knowledge” or “something learned” (“mathematic, n. and adj.” OED
Online).
5
Mac Lane 4.
123
become largely divorced from the latter. Contemporary mathematical platonism is defined
by the core belief that mathematical objects are real, abstract, and exist independent of
agents.6 Thus, for platonists, mathematical objects — such as numbers, sets, functions,
spaces, polyhedra, etc. — only differ from everyday non-mathematical objects in that they
are abstract rather than concrete. The primary advantage of platonism is that a distinct
account of mathematical truth is unnecessary: mathematical propositions are strongly analogous with everyday propositions and attain their truth values in the same way.7 Typically,
platonists hold that the truth value of a proposition derives from whether it accurately reflects the objective facts. It is the abstractness of mathematical objects that distinguishes
mathematics from empirically based subjects. Whereas the physical world is in flux and its
truths are dynamically contingent — the cat was on the mat earlier, but is now hunting birds
in the garden — the abstract objects of mathematics are unchanging, outside of space and
time, so truths about them inherit an unyielding robustness. Moreover, because mathematical objects are abstract, our knowledge of them does not depend on our physical sensations
and experiences, and is therefore a priori. Varieties of platonism have been the foremost
philosophy of mathematics for over two millennia because they seem to provide a simple yet
robust account of mathematics that fits with standard semantics and other traditional ideas
about language use, and with the experiences of practicing mathematicians.
But the primary flaw in the platonist position also has to do with the abstractness of
mathematical objects. First, the principle of parsimony provides a strong reason to consider
theories of mathematics with simpler ontologies; a theory which is able to explain mathematics without recourse to abstract objects will be preferable to platonism. Moreover, the
common understanding of the world predominated by the physical sciences comes with a
tendency to find platonic mathematical objects worryingly mystical. While we have a causal
story explaining how sensory perception of physical objects occurs, there is no parallel explanation of how we come to experience abstract objects; this is one area where physical and
abstract objects are not relevantly similar. This is often referred to as the access problem
6
Øystein Linnebo, “Platonism in the Philosophy of Mathematics”, Stanford Encyclopedia of Philosophy,
2 May 2012. I will adopt the convention of using the uncapitalized “platonism” as an abbreviation for
mathematical theories that possess this core belief.
7
James Robert Brown, Philosophy of Mathematics, 2nd ed. (New York: Routledge, 2008), 12.
124
(or, sometimes, Benacerraf’s problem): how can physical beings come to know abstract objects that lack causal efficacy?8 Plato’s solution to the access problem was to postulate that
learning is recollecting knowledge of the Forms gained during the periods between death and
rebirth when we, like mathematical objects, are disembodied.9 This fantastical explanation
is one of the reasons that most contemporary platonists distance themselves from the traditional theory of Forms. However, no alternative solution to the access problem is widely
agreed upon.
The rejection of platonism necessarily involves a denial of at least one component of its
core conception of mathematical objects: existence, abstractness, or human-independence.
One possibility is to reject abstractness but retain the other two components, resulting in an
extreme and untenable variety of mathematical empiricism where all objects are known only
through sensory perception.10 There is an obvious objection that empiricism must overcome,
which is often presented by platonists as an argument in favour of the abstractness of mathematical objects. It seems impossible that certain mathematical objects could exist in the
physical world. For example, if the universe is understood to consist of finitely many discrete
particles, that makes it difficult to empirically explain the uncountablity of the integers and,
worse still, the infinite continuum of real numbers.11 Moreover, humans perceive only three
spatial dimensions and one temporal dimension, whereas mathematicians routinely work with
spaces and objects with many more than four dimensions; clearly, such mathematical objects
cannot exist within the world as we perceive it. Any contemporary empirical account must
include an answer to this objection among its foundations.
A more telling objection works against the shared realist dimension of platonism and
8
Brown 16–7. Brown does not provide a positive account of how we are acquainted with mathematical
objects, but instead rejects the access problem on the grounds that Einstein-Podolsky-Rosen “spooky action
at a distance” provides a fatal counterexample to the causal theory of knowledge (17–9). In addition to
failing to provide a positive account, it is not clear that Brown’s approach does the work he wants it to given
that “Benacerraf’s problem is remarkably robust under variation of epistemological theory” (Horsten). See
Benacerraf’s “Mathematical Truth” (1973) for the original statement of the objection.
9
See Plato’s Meno and Phaedrus for details.
10
Historically, few authors have defended any form of mathematical empiricism (John Stuart Mill is the
canonical example), and none of those few has taken this naive hard line, as far as I am aware. However,
the contemporary scientific paradigm has given rise to a new interest in more sophisticated varieties of
mathematical empiricism. For example, Philip Kitcher describes his approach in The Nature of Mathematical
Knowledge as empirical (4).
11
Brown, Philosophy of Mathematics 13–4.
125
empiricism. Mathematical realists believe that independently existing mathematical objects
are necessary to ground objective mathematical truths. Stephen Yablo’s concise objection
to realism takes the form of a dilemma: either our conception of numbers is determinate or
it is not.12 If our number concept is determinate, then that conception alone is sufficient
for the objectivity of truths about numbers whether it actually refers to number objects
or not, making the objects redundant. If our number concept is not determinate then an
explanation of how it successfully refers to an objectivity-providing structure is required.
Yablo contends that this can only be done if only one structure in the extension of our
conception exists. However, this scenario admits the possibility of Putnamian Twin Earth
cases where a person internally identical to me makes an arithmetical statement that would be
true coming from my mouth (“7 is a prime number,” for example), but is false on Twin Earth
because it refers to twintegers that differ from the integers. Hence, our arithmetical concepts
are revealed to be externalist and the characteristic necessity of mathematical statements is
compromised, entailments that are generally held to be unacceptable by realist philosophers.13
Such general arguments against mathematical realism in concert with the specific difficulties
facing platonism and empiricism provide motivation to consider other — and, particularly,
antirealist — positions.14
Two related advents in mathematics in the late nineteenth century allow for the development of serious alternatives to platonism. The first was a bevy of improvements in formal
logic, set theory, and axiomatics. The second was the development of non-Euclidean geometries. In particular, the latter provided evidence against the platonistic idea that there
is only one true mathematics. However, the former is directly responsible for one of the
first alternative approaches in the philosophy of mathematics. Logicism is defined by the
belief that mathematics is, in some nontrivial sense, reducible to logic. This usually means
12
For Yablo, a determinate conception is one for which any statement made about an exemplar of the
concept is unambiguously and decidedly true or false (“Go Figure” 194).
13
Yablo, “Go Figure” 194–5.
14
The taxonomy of mathematical theories is not as clear as the discussion in this section makes it out
to be; for example, there may be positions self-identifying as varieties of platonism or empiricism that are
less committed to the independence of mathematical objects. Likewise, not all of the positions considered
below are necessarily incompatible with realist assumptions. For example, Frege is often identified as both a
platonist and a logicist. The reader is once again reminded that this overview of the philosophy of mathematics
is drastically simplified and generalized by necessity.
126
attempting to show that all mathematical truths are tautological, logically derivative from
a set of self-evident axioms. Accomplishing this would establish an ontology-independent
foundation for mathematical knowledge; for example, “All polytopes are polytopes” is a tautology regardless of whether polytopes exist. The emphasis in logicism is on proof and the
relationships between concepts. Gottlob Frege is generally recognized as the first logicist,
though he drew inspiration from Leibniz’s conjecture that arithmetic is reducible to logic.15
The revolutionary logical developments contained in his Begriffsschrift allowed Frege to make
the first concerted attempt to describe arithmetic as a formal deductive system in his twovolume Grundgesetze der Arithmetik, an undertaking that previous logics were ill equipped
to handle. However, a major flaw in Frege’s system was exposed by Bertrand Russell just
as the second volume of the Grundgesetze was going to press, namely that Frege’s Basic
Law V entails Russell’s paradox and, therefore, is inconsistent.16 In order to circumvent the
paradox, Russell developed his theory of types, an alternative to naive set theory which restricts problematic self-referentiality. This laid the groundwork for Whitehead and Russell’s
Principia Mathematica, arguably the pinnacle of logicist achievements and one of the most
significant publications of the twentieth century. Though it is less read today, the influence
of the Principia permeates contemporary mathematics.
While logic is undeniably related to mathematics, logicism fails to be an adequate philosophy of mathematics. Before looking at the criticisms, some positive contributions of logicism
should be noted. Logicist ideals have been a part of mathematics since Euclid: every mathematician aspires to be as systematic, clear, and rigorous as the Elements. Logicism takes
this impulse to its natural conclusion, attempting to expose the entirety of mathematics as a
single axiomatic system. In trying to achieve this goal, logicists not only generated a variety
of important fundamental mathematical results, but changed the way mathematics was practiced by implementing new techniques and standards for proofs. Improved symbolization and
other developments in formal logic also changed the way philosophy was practiced, resolving some long standing problems and giving rise to new questions. The reductive successes
achieved by the logicists paved the way for the development of the computer, which utilizes
15
16
Michael D. Resnik, Frege and the Philosophy of Mathematics (Ithaca: Cornell UP, 1980), 13.
Resnik 211–2.
127
electronic logical circuitry to perform arithmetic calculations; questions concerning the nature of mathematics qua logical system (specifically, the Entscheidungsproblem) motivated
the creation of the theoretical Turing machine.17 Despite its mathematical and philosophical
successes, logicism is untenable as a foundational philosophy of mathematics. First, it is not
possible to reduce mathematics to logic without including set theory (or some equivalent);
only absolute purists would claim that the logicist enterprise fails because of this.18 However, even more forgiving scholars acknowledge that mathematics requires certain axioms
and inferential rules that are distinctly non-logical, including the axiom of infinity and the
principle of mathematical induction.19 Moreover, there are some mathematical statements
— the continuum hypothesis is the canonical example — which can be neither proved nor
disproved from the axioms of set theory and which, therefore, must stand alone as independent axioms.20 Another criticism is that logicism takes for granted rather than shows that
logic itself is certain and reliable; thus, even if the reductions were successful, it would remain unclear that the foundations of mathematics are secure.21 Third, logicism has difficulty
explaining the acquisition and development of mathematics, which does not begin with the
axioms. While much of mathematics may be generated from a small collection of axioms,
nobody learns or performs arithmetical calculations in the manner presented in the Principia.
Logicism, therefore, does not accommodate a wide range of our mathematical experiences.22
Arguably, the most devastating blow to logicism comes from Gödel’s incompleteness results; I
shall postpone discussion of this fatal criticism briefly to introduce formalism, a philosophical
sibling of logicism equally shaken by that objection.
From some perspectives, the distinction between logicism and formalism seems slight:
both viewpoints seek foundational certainty in the conception of mathematics as a formal
axiomatic system rather than in objective abstract objects. Where they differ is on their
17
Alan Turing, “On Computable Numbers, with an Application to the Entscheidungsproblem,”Proceedings
of the London Mathematical Society, Series 2, 42 (1937): 230–65.
18
Paul Benacerraf and Hillary Putnam, “Introduction,” Philosophy of Mathematics: Selected Readings,
Eds. Benacerraf and Putnam (Englewood Cliffs: Prentice-Hall, 1964), 10.
19
It is noteworthy that every example of a non-logical mathematical axiom I have encountered involves
infinity in some way.
20
Paul Ernest, Social Constructivism as a Philosophy of Mathematics (Albany: SUNY Press, 1998), 16.
21
Ernest 17–8.
22
This line of criticism is similar to one presented by Wittgenstein in Remarks on the Foundations of
Mathematics, Part II (1956).
128
understanding of mathematical truth. Recall that for the logicist, mathematical truths are
logical truths. On the other hand, formalists hold that the formal system of mathematics is
purely syntactic, the result of the repeated application of sanctioned symbol-manipulation
rules to an ever expanding collection of symbolic strings.23 These symbolic strings are thus
devoid of meaning unless some interpretation is imposed on them.24 For the formalist, then,
the key issue is not establishing that mathematics is true in the conventional sense but rather
that the formal system is consistent, free from paradox and contradiction. Formalists have
an advantage over logicists in that they are free to use non-logical or seemingly ad hoc rules
and axioms so long as they are consistent with their system. On the other hand, a common
complaint about formalism is that it seems ill equipped to explain why the meaningless game
of mathematics should be so “unreasonably effective” (as Wigner so aptly put it), or why
mathematics ought to be privileged over any other meaningless game.25 However telling this
criticism may be, the foundational ambitions of both logicism and formalism were effectively
terminated by one of the most astounding intellectual results of the twentieth century.
In 1900, David Hilbert, one of the foremost mathematicians of his day and noted formalist,
presented a collection of 23 significant open problems to the mathematical community. The
second of these problems asked for a proof of the consistency of the axioms of arithmetic
(that is, that the system derived from those axioms does not contain a contradiction).26
Kurt Gödel’s graduate research aimed to provide such a proof, but instead culminated in
the publication of his infamous incompleteness theorems in 1931. These results can be
23
Another way of putting this is that logicists thought mathematics had a null subject matter and “dealt
with pure relations among concepts” whereas formalists thought that the subject matter of mathematics
was comprised of symbolic expressions (Benacerraf and Putnam 9). Note that no account of metaphor takes
it to be purely syntactic, and thus the intuition that metaphor plays a constitutive role in mathematics is
incompatible with formalism.
24
David Hilbert, the father of formalism, contended that mathematics was composed of two types of
formulae: “first, those to which the meaningful communications of finitary statements correspond; and,
secondly, other formulas which signify nothing and which are the ideal structures of our theory” (“On the
Infinite” 146). That is, it seems that Hilbert thought that elementary arithmetical statements involving finite
integer quantities can be certain perceptual truths (as Kant did), but that statements involving infinity are
meaningless but instrumentally useful extensions of the finitary (Brown 70–1). This provides a reminder that
formalism — and, indeed, each of the viewpoints considered — is more complicated than the brief sketch I
present here.
25
Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications in Pure and Applied Mathematics 13.1 (1960): 1.
26
David Hilbert, “Mathematical Problems,” Bulletin of the American Mathematical Society 8.10 (1902):
447.
129
paraphrased non-formally as:
1. Any consistent formal system containing ordinary arithmetic is incomplete (contains
a sentence such that both the sentence and its negation are unprovable within the
system).
2. It is not possible to prove the consistency of such a formal system within that system.27
It is the second of these two results that deals the fatal blow to projects seeking to establish certain axiomatic foundations for mathematics; as Paul Ernest puts it “[t]he second
incompleteness theorem showed that in the desired cases consistency proofs require a metamathematics more powerful than the system to be safeguarded; thus there is no safeguard
at all.”28 While Gödel’s incompleteness theorems show that the formalism cannot provide
robust philosophical foundations for mathematics, they do not invalidate the use of formal
axiomatic and reductive set-theoretic approaches by mathematicians. Indeed, from a mathematical perspective, formalism can be seen as a great triumph insofar as it brought increased
rigor, clarity, and unity to the subject and led to a number of fundamental and fascinating
results — including Gödel’s theorems themselves! The failure of the attempt to reduce mathematics to its formal aspect does not imply that this aspect is eliminable; one must remain
cognizant of mathematics’ game-like and symbol-dependent character in theorizing about it.
Two more perspectives warrant attention to round out the philosophical spectrum, though
they traditionally have had fewer adherents than the other positions discussed above. Both
of these positions are classified as versions of constructivism, as they share the conviction
that mathematics is a human-dependent construction.29 The first of these positions, intuitionism, arose at roughly the same time as logicism and formalism, and completes the
list of the traditional alternatives to platonism. The central commitment of intuitionism is
27
Mac Lane 379. Note that Gödel and others have proven the consistency of weaker formal systems,
including the sentential and predicate calculi.
28
Ernest 19. While there is widespread agreement on this point, a few philosophers (notably, Michael
Detlefsen) contend that Gödel’s incompleteness results do not constitute a definitive answer to Hilbert’s
second question (Detlefsen 345–6). However, given that logicists and formalists ultimately wish to consider
the entirety of mathematics as a single axiomatic system, Gödel’s result seems to entail that a consistency
proof would have to be non-mathematical, an unpalatable consequence for most.
29
One may argue that “human-dependent” is too strong a condition, as it rules out the possibility of
intelligent non-human entities doing mathematics.
130
that “mathematics is an essentially languageless activity of the mind having its origin in the
perception of a move of time.”30 That is, our capacity to distinguish successive moments
in time combined with our powers of abstraction gives rise to the integers.31 Grounding
arithmetic upon our time sense leads the intuitionists to some unusual conclusions. The fact
that many statements about the future seem to be neither true nor false but indeterminate
helps motivate the intuitionist rejection of the law of the excluded middle. This means that
intuitionists are not allowed to use proofs by contradiction in their mathematics.32 Because
there is always room for more moments in the future but those future moments do not yet
exist, intuitionists accept the idea of a potential infinity but reject absolute infinities.33 These
rejections mean that an intuitionist mathematics will have a restricted scope in comparison to
mathematics associated with other philosophical viewpoints. The primary complaint against
intuitionism is that these restrictions entail the unacceptable loss of many important results
of classical mathematics, some of which play a pivotal role in our most powerful scientific
theories.34 Another criticism is that the central commitment of intuitionism makes mathematics mind dependent and leads to a problematic relativism; some possible constructivist
responses against charges of psychologism will be considered later in the chapter. While most
see these objections as sufficent grounds for rejecting intuitionism, some devoted individuals
continue to do mathematics according to intuitionist assumptions. Most mathematicians
will agree that constructive proofs are powerful and can give insights which non-constructive
proofs may fail to.35 However, this merely provides a reason to favour constructive proofs
30
L.E.J. Brouwer, Brouwer’s Cambridge Lectures on Intuitionism, Ed. D. van Dalen (Cambridge: Cambridge UP, 1981), 4–5.
31
In his Critique of Pure Reason, Immanuel Kant theorizes that mathematical knowledge emanates from
the pure intuitions of space and time that provide a priori form to our cognitions. Thus, intuitionism borrows
both its central commitment and its name from Kant’s philosophy of mathematics (Brown 119–20). While
Kant’s theory of mathematics is historically significant and influenced Frege, Hilbert, and Brouwer, it has
almost entirely fallen out of favour.
32
John D. Barrow, Pi in the Sky: Counting, Thinking, and Being (Oxford: Clarendon, 1992), 185–6.
33
Brown 128.
34
Brown 134.
35
A non-intuitionist may demonstrate the existence of some mathematical object by proving the assumption
of its non-existence leads to contradiction; such non-constructive proofs provide little insight into how one
might arrive at an instance of such an object. A constructive proof, on the other hand, demonstrates existence
by providing a direct method for constructing an explicit instance of the object in question. The Intermediate
Value Theorem (If f is continuous on the interval [a, b] and there is a C with f (a) < C < f (b) then there
exists a c such that a < c < b and f (c) = C) can only be proved non-constructively: there is no way to prove
the theorem by showing a method for finding c directly (Brown 131). Brown notes that not all constructive
131
whenever they are possible, not to rely on them exclusively.
If intuitionism is rooted in psychology, then the second constructivist position considered
here, social constructivism, is sociological in nature. Social constructivist theories hold that
mathematics is a cultural institution, though their commitment to this varies in degree.
The weakest versions of social constructivism involve only an acknowledgment that at least
some mathematical concepts have historical and cultural aspects, a generally noncontroversial
thesis. The growing literature on the history of mathematics provides significant evidence in
favour of weak social constructivism.36 Indeed, one need look no further than intuitionists
and non-intuitionists to find two groups who practice mathematics in different ways! On
the other end of the spectrum, the strongest social constructivisms hold that mathematics
exists solely by social convention. This extreme end of the spectrum is populated primarily by
straw men, as it seems to allow the possibility that arbitrary assumptions are mathematically
legitimate as long as they are backed by the consensus of some community. Thus, both
intuitionism and social constructivism need to answer charges of relativism. However, on
a positive note, these constructivist philosophies make some important observations about
mathematics-as-practice, usefully shifting the traditional focus.
All of the above viewpoints have been been deemed unacceptably flawed. However, each of
them captures something important about contemporary mathematical practice. Mathematicians talk and act as though mathematical objects have an objective platonistic existence;
while it seems to be theoretically possible to alter our language to eliminate reference to
mathematical entities, implementing such a change is thoroughly impractical and unnecessary. Much mathematical work is highly symbolic and formal, and is conceived of as being
part of an axiomatic system. And yet, mathematical progress is clearly not a straightforward
linear unpacking of axiomatic implications but rather involves hunches, hypotheses, and trial
and error, and only becomes rigorous and formalized at a later stage. What seems to be
proofs provide direct methods; for example, one may constructively prove the compositeness of an integer
using Fermat’s theorem without giving a prime factorization or indicating how we might arrive at one (Brown
133).
36
Imre Lakatos played a key role in the popularization of social constructivism. His Proofs and Refutations (1976) takes the form of a dialogue that mirrors the historical developments surrounding the concept
POLYHEDRON. This work opposes the tradition by arguing that definitions should be seen not as the starting
point of mathematical practice but rather the end result of a process of conceptual development.
132
required is a philosophical position that acknowledges and incorporates the various aspects
of mathematical practice rather than detrimentally focusing on one exclusively. While years
of disagreement between the traditional viewpoints have exposed apparent incompatibilities
that have been seen as an impediment to an amalgamated perspective, metaphor provides a
promising mechanism for attaining an integrated theory.
If metaphor is to do the desired work, the following two preconditions must hold true.
First, the understanding of metaphor invoked will necessarily be a non-traditional one, as
traditional theories hold that metaphors are eliminable. Second, any theory of mathematics
that includes metaphor as a constitutive element must acknowledge that mathematics must
be at least partially human-dependent, to the extent that metaphors are human-dependent.
As has been discussed above, these two ideas have been around for several hundred years but
were not significantly developed until the twentieth century, and they remain controversial
today. This helps explain why few authors have explored the possibility of metaphorical
explanations of mathematics, and why such views did not emerge until recently. I consider
two such theories below. Lakoff and Núñez’s CMT-based theory of embodied mathematics
is arguably the most ambitious and successful attempt to include metaphor as a constitutive
element in a philosophy of mathematics. Their work is very promising, but it is still in its
early stages and is not beyond criticism and improvement. Yablo’s mathematical figuralism
takes a different approach to incorporating metaphor, one that involves the idea of fictional
objects. Yablo’s theory initially provides contrast to Lakoff and Núñez’s, but I ultimately
argue that there is an underlying compatibility between Lakovian embodied mathematics
and certain varieties of mathematical fictionalism.
4.2
Conceptual Metaphor Theory and Embodied Mathematics
Thanks to CMT’s focus on the conceptual, discussion of mathematics has been a part of the
project since its inception.37 However, it is not until 20 years into the CMT project that
37
See, for example, Metaphors We Live By, pages 218–22, and Women, Fire, and Dangerous Things,
chapter 20.
133
mathematics is given a thorough treatment in Where Mathematics Comes From. This book
begins with the question “Exactly what mechanisms of the human brain and mind allow human beings to formulate mathematical ideas and reason mathematically?”38 By taking this
as the central question of their inquiry, Lakoff and Núñez invert the traditional approach,
which demands solid objective mathematical foundations before complicating matters with
considerations of fallible and idiosyncratic human experience and behaviour. Explaining the
role of the brain in observed mathematical practices is an empirical undertaking, not a matter of a priori philosophical or mathematical theorizing. However, while Where Mathematics
Comes From is primarily a work in theoretical cognitive science, the authors do also explicitly explore many immediate philosophical consequences of their research. In particular,
they require their theory to account for the precision, consistency, stability, cross-cultural
understandability, symbolizability, calculability, and descriptive and predictive effectiveness
of mathematics.39 My primary interest is not in whether the speculative details of their
“mathematical idea analysis” are accurate but in the ramifications of their approach in general; thus, I will limit discussion of specific conceptual metaphors below, invoking them only
as necessary to support the metaphorical approach generally.
Lakoff and Núñez’s account of mathematical concepts is founded squarely upon CMT:
worldly perceptual stimuli interact with our genetically determined mental and bodily structures and capacities to yield direct experiences. Correlations between these direct experiences
give rise to the most-basic conceptual metaphors which allow us to understand one experience
in terms of another, thereby producing a new level of indirect experiences. These resultant
experiences are then available as inputs for further metaphorical mappings and thus the cycle continues, increasing abstraction and complexity of both experience and metaphor as the
layers stack. It is worth considering each of these stages in more detail. First, the biological
capacities underpinning mathematical reasoning. There is a growing body of biological evidence regarding numerical competency. Studies targeting the capabilities of human infants
and various non-human animal species are of particular interest as they provide evidence that
some numerical abilities are ancient and genetically hardwired. Experiments indicate that
38
39
Lakoff and Núñez, Where Mathematics Comes From 1.
Lakoff and Núñez, Where Mathematics Comes From 50.
134
a wide range of animals have rudimentary arithmetical abilities. It is perhaps unsurprising
that chimpanzees possess a fairly wide range of numerical capabilities, from discriminating
the sizes of small collections of objects to arithmetic with small integers and simple fractions.40 Somewhat more surprising is the fact that there is evidence that many mammals
and birds — including rats, cats, raccoons, lions, dolphins, parrots, and pigeons — possess
similar capabilities when it comes to small integers.41 It has been shown that angelfish are
able to reliably discriminate quantities up to 4, showing that numerical capabilities are not
confined to higher vertebrates.42 And, astoundingly, research involving bees and mealworms
suggests that numerical discrimination abilities extend even to invertebrates.43 The ability to
numerically discriminate seems clearly advantageous from an evolutionary standpoint insofar
as it helps individuals optimize outcomes when confronted with a choice between differently
sized food sources, breeding groups, or defense organizations. This has led Dehaene to hypothesize the existence of a genetically determined “number sense” that explains the above
animal results.44
Experiments suggest that this number sense also exists in infant humans, lending credence
to the idea that these capabilities are genetically determined.45 Prior to 1980, the majority
of psychologists subscribed to the Piagetian view that every newborn is a tabula rasa, mathematically speaking.46 The earliest studies investigating the numerical capacities of babies
contradicted this viewpoint by demonstrating that infants are able to discriminate between
visual representations of 2 and 3 a few days after birth.47 These experiments involved repeatedly showing infants slides with two objects of varying location, size, and identity on
them until habituation was achieved, then presenting a slide with three objects. The atten40
Stanislas Dehaene, The Number Sense: How the Mind Creates Mathematics, 2nd ed. (New York: Oxford
UP, 2011), 14.
41
Lakoff and Núñez, Where Mathematics Comes From 21 and Dehaene, Chapter 1. Note that the ability
to add fractions has only been observed in primates to date (Dehaene 14).
42
Luiz M. Gómez-Laplaza and Robert Gerlai, “Spontaneous discrimination of small quantities: shoaling
preferences in angelfish (Pterophyllum scalare),” Animal Cognition 14.4 (2011): 565–74.
43
See Dacke and Srinivasan (2008) and Carazo et al. (2009).
44
Dehaene 28.
45
Lakoff and Núñez refer to these inborn abilities as “innate arithmetic” (Where Mathematics Comes From
19). I am reluctant to adopt this terminology, as the word “innate” has a variety of philosophical connotations
that I expressly do not wish to invoke.
46
Dehaene 31.
47
Lakoff and Núñez, Where Mathematics Comes From 15.
135
tion fixation duration of the infants was significantly longer after the change, indicating a
discrimination between the two stimuli that was attributed to a sensitivity to numerosity.48
Subsequent experiments showed that newborns can likewise discriminate between auditory
stimuli consisting of two and three syllables and, more significantly, that when six-month
olds are simultaneously presented with one sequence of tones and two visual representations
of different quantities, they consistently attend longer to the slide whose numerosity matches
the auditory stimulus. This latter study provides strong evidence that our inborn sensitivity
to numerosity is abstract and amodal rather than tied to visual or auditory perception, that
is, “that the child really perceives numbers rather than auditory patterns or geometrical
configurations of objects.”49
The numerical competency of preverbal infants is not limited to mere quantity discrimination, but also seems to include some rudimentary arithmetic. In the early 1990s, it was shown
that 4- and 5-month old babies possess some understanding of the addition and subtraction
of small numbers. Researchers presented the infants with a scene where two distinct objects
were placed behind a screen and then the screen was dropped, revealing either one, two, or
three objects. It was observed that babies attended significantly longer to the two impossible
scenarios than to the possible one, leading to the conclusion that the infants had expected
to see two objects and were surprised or puzzled by the outcomes that conflicted with this
expectation.50 These studies and others show that infants possess arithmetic capabilities far
beyond the expectations of previous generations of researchers. Though these results are
significant and amazing, the extreme limitations of infant mathematics must be stressed. Infant arithmetic does not seem to extend beyond cardinality recognition, sums, and differences
involving the integers between 1 and 4, and, significantly, does not even include an understanding of an ordering on those integers.51 These capabilities are more limited than those
of adult chimpanzees, let alone adult humans.52 It seems our inborn powers are insufficient
48
Dehaene 38.
Dehaene 40. Though he rejects the view, Dehaene concedes that “[w]hile waiting for conclusive experiments with younger children, it remains possible to maintain that learning, rather than brain maturation, is
responsible for the baby’s knowledge of numerical correspondence between sensory modalities” (50).
50
Dehaene 42.
51
Dehaene 51.
52
Dehaene 45.
49
136
to handle arithmetic with large integers, let alone transfinite number theory, tensor calculus,
and the rest of the gamut of human mathematics. Experiences and bodily developments
must make a contribution.
It is clear that, as they age and develop, humans eventually acquire a wide range of
mathematical knowledge and skills, most of which are connected to their capacity for symbol
and language use, and the instruction they are thereby able to receive from their parents and
teachers. A relevant question that arises is this: what is the connexion between inborn infant
mathematics and symbolic adult mathematics? Some, such as extreme social constructivists,
may hold that any correspondences between infant arithmetic and adult mathematics within
contemporary individuals are coincidental, and claim that the two capacities develop entirely
independently of each other. Most people, however, would likely be inclined to posit a
parsimonious connexion between the two. Some evidence in favour of such a position comes in
the form of the human subitizing capacity — our ability to quickly and accurately discern the
cardinality of small collections of discrete objects (up to about four), a capability which seems
to differ from counting (understood as sequential pairing).53 A controversial but plausible
hypothesis is that subitizing in adults and quantity discrimination in infants utilize the
same neural circuitry.54 As one might expect, Lakoff and Núñez believe that there is a
strong connexion between infant and adult mathematics, endorse the above hypothesis and,
further, claim that cognitive mappings — particularly, though not exclusively, conceptual
metaphor — provide the mechanism by which our sparse inborn arithmetical capabilities,
image schemata, and various other protoconceptual mechanisms bring about the full range
of human mathematics.55
Lakoff and Núñez introduce a classification scheme for mathematical conceptual metaphors
in Where Mathematics Comes From that differs from the one I presented in chapter 3. Their
primary distinction is between grounding and linking metaphors. Grounding metaphors map
inferences about everyday experiences onto abstract concepts, connecting our inborn arith-
53
Lakoff and Núñez, Where Mathematics Comes From 19.
Dehaene 57.
55
Recall, from chapter 3, that an image schema is a cognitive mechanism that interactively imposes form
on perceptions, thereby providing protoconceptual structure that facilitates basic reasoning and agency in
the physical world (Lakoff and Núñez, Where Mathematics Comes From 30–1).
54
137
metic capacities to our basic agential interactions with the world, for example. It seems clear
that there will be some overlap between the class of grounding metaphors and the class of
primary metaphors discussed in chapter 3. Linking metaphors bridge two branches of mathematics, forming links between abstract concepts.56 Linking metaphors that privilege their
source domain as fundamental form the subclass of foundational metaphors; such mappings
lurk at the core of attempts to formally reduce one branch of mathematics to another.57 Extraneous metaphors are those which “have nothing whatever to do with either the grounding
of mathematics or the structure of mathematics itself . . . [they] can be eliminated without
any substantive change in the conceptual structure of mathematics.”58 While they may aid
visualization or reference, extraneous metaphors do not make a constitutive contribution to
mathematics.59 This classification is important insofar as it is a key structural element in
Lakoff and Núñez’s writing and is the locus of some important criticisms of their theory. It
is important to note that while there is a strong emphasis on conceptual metaphor in Where
Mathematics Comes From, several other cognitive capacities also play constitutive roles in
their theory — including, but not limited to, conceptual metonymy, conceptual-blending,
and symbolization.60
Lakoff and Núñez posit that four fundamental grounding metaphors work in concert, using
our direct worldly experiences to enrich our inborn, pre-conceptual quantity discrimination
abilities into a
NUMBER
concept capable of supporting arithmetic in its full richness. Recall
the hypothesis from chapter 3 that primary metaphors arise through neural conflations, “the
simultaneous activation of two distinct areas of our brains, each concerned with distinct as56
Lakoff and Núñez, Where Mathematics Comes From 52–3.
Lakoff and Núñez, Where Mathematics Comes From 100.
58
Lakoff and Núñez, Where Mathematics Comes From 52.
59
Mathematical language is filled with eliminable catachreses: the legs of a right triangle, rings, sheaves,
wreath products, stair functions, telescoping sums, and so forth. The object of study in my M.Sc. thesis
is the pinwheel tiling, a mathematical object so named because its triangular tiles occur in an uncountable
number of rotational configurations as do the blades of the rotating children’s toy. Such metaphors typically
do not penetrate deeper than the names and therefore do not infect the rigorous mathematical reasoning
with their imprecision: no one thinks of the vertices of a right triangle as hips or feet!
60
These other cognitive elements are often overlooked and deserve more attention than Lakoff and Núñez
have given them. For example, the Fundamental Metonymy of Algebra allows roles played to stand for
individuals and is hypothesized to be the mechanism which “allows us to go from concrete (case by case)
arithmetic to general algebraic thinking” (Lakoff and Núñez, Where Mathematics Comes From 74). Clearly,
such an important abstracting mechanism warrants more than a mere half a page of discussion! Rectifying
such deficiencies falls outside the scope of this dissertation on metaphor and mathematics.
57
138
pects of our experience.”61 The first of the arithmetic grounding metaphors,
OBJECT COLLECTION,
ARITHMETIC IS
arises from regular and frequent correlations in our experience between
perception and manipulation of groups of physical objects and basic quantity discrimination,
addition, and subtraction; the multimodality of our inborn number sense provides numerical experiences not associated with any object collections that allows the source and target
domains to be differentiated.62 It is hypothesized that
ARITHMETIC IS OBJECT COLLECTION
is
established at a very young age, long before any mathematical training occurs.63 Human
experiences of collections of objects possess extensive image-schematic structure that facilitate our agential interactions with those collections. For example, a constitutive part of
our protoconceptual understanding of collections of physical objects is that the introduction
of a new object into a collection A results in a different collection B, not a different kind of
thing altogether (a non-collection).64 Because conceptual metaphors are inference preserving,
the grounding metaphor
ARITHMETIC IS OBJECT COLLECTION
is able to enrich our paltry inborn
arithmetic by importing reasoning structures pertaining to physical collections. The preservation of collectionhood under object introduction maps to the preservation of numberhood
under addition, for example, giving rise to part of the mechanism of mathematical closure.
The metaphorically induced mechanism of mathematical closure is necessary to extend
our arithmetic beyond the first three integers and, additionally, provides one of the clearest
illustrations of conceptual metaphors creating target domain entities. Infants’ distinct experiences of 1, 2, and 3 and the connections between them, such as 2 + 1 = 3 and 2 − 1 6= 2,
constitute a stable, structured protoconceptual understanding of numbers. The metaphor
ARITHMETIC IS OBJECT COLLECTION
induces further distinct numerical experiences that extend
61
Lakoff and Núñez, Where Mathematics Comes From 42.
Whereas our “affection sense” is only triggered by a fairly small set of relatively specific circumstances
(being cuddled, for example), our number sense is almost constantly working; this is because these two
mechanisms work quite differently in service of our evolutionary fitness. Thus, AFFECTION and WARMTH
are experienced independently more often than NUMBER and OBJECT COLLECTION, so the disanalogy is better
established in the former case; that is, the source and target domains are recognized as connected, but
distinct. I speculate that the conflationary coactivations involved in ARITHMETIC IS OBJECT COLLECTION are so
persistent that differentiation between these domains is typically incomplete, leading to the impression that
the relationship between them is something stronger than metaphor.
63
Lakoff and Núñez, Where Mathematics Comes From 54–5.
64
Lakoff and Núñez, Where Mathematics Comes From 81. Notice the use of the word “into” in the above
sentence exposes part of the image schematic structure involved; see page 39 of Where Mathematics Comes
From for more discussion of the INTO schema.
62
139
our understanding to include the other positive integers. The reasoning seems go something
like this: a baby can recognize 1 and 2 as numbers and 3 — the result of 1+2 — as a number,
and also can recognize that introducing one more grape to an existing handful of three grapes
changes the collection of grapes. Thus, because they distinguish 1 and 3 as numbers, the
metaphor maps the introduction of a grape to the sum 3 + 1, and the outcome to a new,
different number: 4. Two important qualifying comments are necessary. First, despite the
descriptions contained in the previous two sentences, one must remember that a large portion
of this development is hypothesized to take place in the first few months of life, long before
the acquisition of symbolic numerals.65 Additionally, the development of this metaphor does
not occur in a void, but rather seems to require the assistance of “appropriate language, emotional support, and behaviour.”66 Second, it should be made clear that, even in newborns,
the estimating faculty that allows us to distinguish between groups of objects is active when
presented with collections of any cardinality, including those larger than three; however, we
do not have enough inborn structure to consistently, precisely, and systematically distinguish
between experiences of larger groups at birth.67 The grounding metaphor bestows these
properties upon the larger integers by transferring structure derived from our understanding
of collections of physical objects that extends the consistent, discrete structure of the small
integers to our larger quantitative approximations.
ARITHMETIC IS OBJECT COLLECTION
creates
the number zero once the understanding of collections is expanded to include the idea of an
empty collection, the result of taking one object away from a collection of one object, for
example.68 Lakoff and Núñez also claim that
ARITHMETIC IS OBJECT COLLECTION
is responsible
for imposing an ordering on the integers that comes from an ability to distinguish which of
two collections is bigger, a cognitive mechanism with clear survival benefits.69 The other
three arithmetic grounding metaphors —
ARITHMETIC IS OBJECT CONSTRUCTION, ARITHMETIC IS
65
Lakoff and Núñez, Where Mathematics Comes From 55.
Rafael Núñez, “Numbers and arithmetic: neither hardwired nor out there,” Biological Theory 4.1 (2009):
78. Dehaene suggests that studies of the (until recently) isolated Mundurukú and Pirahã peoples show that
gaps in this cultural scaffolding may result in an incomplete acquisition of ARITHMETIC IS OBJECT COLLECTION
(263).
67
Lakoff and Núñez, Where Mathematics Comes From 51. Lakoff and Núñez refer to this ability to estimate
quantities as the numerosity capacity.
68
Lakoff and Núñez, Where Mathematics Comes From 64.
69
Lakoff and Núñez, Where Mathematics Comes From 56.
66
140
USE OF A MEASURING STICK,
and
ARITHMETIC IS MOTION ALONG A PATH
ARITHMETIC IS OBJECT COLLECTION
— systematically extend
in turn, allowing for the creation of fractional, irrational,
and negative numbers respectively. These four are not the only grounding metaphors for the
entirety of mathematics; one other example discussed in the book is
CLASSES ARE CONTAINERS
as a grounding metaphor in set theory.70
Whereas grounding metaphors have extensively structured sensorimotor source domains,
linking metaphors are not directly grounded in this way: they are maps between two abstract
mathematical domains. Linking metaphors not only allow modeling to occur between previously established branches of mathematics, but can even create entire branches of mathematics, such as trigonometry.71 The majority of the conceptual metaphors presented in Where
Mathematics Comes From are linking metaphors.72 One key example of a linking metaphor is
NUMBERS ARE POINTS ON A LINE,
which connects arithmetic and geometry. Descartes’ develop-
ment of this metaphor blended the two domains together and laid the necessary foundations
for the development of calculus.73 Whereas grounding metaphors are typically acquired automatically early in life thanks to our genetic and social programming, linking metaphors arise
through creative effort and require significant instruction if they are to be attained; because
of this, there are many mathematical linking metaphors that only trained mathematicians
possess.74 Likewise, not every culture possesses the same linking metaphors; for example, our
pre-Cartesian ancestors did not possess
NUMBERS ARE POINTS ON A LINE,
and, apparently, nei-
ther do the Yupno people of Papua New Guinea.75 Linking metaphors are a crucial “part of
70
Lakoff and Núñez, Where Mathematics Comes From 123.
Lakoff and Núñez, Where Mathematics Comes From 150. Discussion of the linking metaphors giving rise
to trigonometry can be found in their Case Study 1, pages 383–98.
72
Lakoff and Núñez, Where Mathematics Comes From 53.
73
Recall from chapter 3 that a conceptual blend, as conceived by Fauconnier and Turner, involves the
superimposition of conceptual structures connected by fixed correspondences, resulting in the creation of
hybrid structures (Fauconnier and Turner 47). A metaphorical blend is a conceptual blend where a conceptual
metaphor constitutes the fixed correspondences. Though Lakoff and Núñez allege that both metaphorical and
non-metaphorical blends play an important role in mathematics, they provide no explicit non-metaphorical
examples (Lakoff and Núñez, Where Mathematics Comes From 48). For an interesting discussion of the
complementarity of conceptual blending theory and CMT, see Grady, Oakley, and Coulson (1999).
74
For example, a person requires at least a basic understanding of mathematical group theory in order to
possess the ROTATION GROUP METAPHOR which maps group elements to geometric transformations (Lakoff and
Núñez, Where Mathematics Comes From 116).
75
Rafael Núñez, Kensey Cooperrider, and Jürg Wassmann, “Number concepts without number lines in
an indigenous group of Papua New Guinea,” PLoS ONE 7.4 (2012): 8. While neither of these example
cultures have a full-fledged number line concept because they are lacking the linking metaphor, Núñez’s
71
141
the fabric of mathematics” insofar as they are responsible for making mathematics coherently
unified across its several branches.76
One other specific metaphor requires attention here. The collaboration between Lakoff
and Núñez was initiated because Núñez’s preliminary research on
INFINITY
led him to hy-
pothesize that metaphor may play a necessary role in conceptualizing it.77 While Where
Mathematics Comes From turned out to be far more than a discussion of
INFINITY,
it does
remain a central topic of the book. Lakoff and Núñez claim that a single metaphor, the
Basic Metaphor of Infinity (BMI), provides the basis for all mathematical conceptions of
infinity. This metaphor is rooted in the aspectual system we use in conceptualizing events
or processes. In particular, the BMI maps completed iterative actions (that is, actions with
a perfective aspect) to indefinitely perpetual iterative actions (actions with an imperfective
aspect). The main effect of this mapping is to transfer the idea of the termination of a process
resulting in a unique final state from the perfective domain into the imperfective domain; in
the latter, actions conceivably go on forever so there is no corresponding structure to map
on to. Thus, the BMI creates actual infinity as the metaphorical end result of a potentially
infinite process.78 Neither the source nor the target domain of the BMI is inherently numerical; to obtain the various mathematical versions of infinity (for example, the final counting
number or the edge of the Euclidean plane) the target domain of the BMI must be restricted
to a specific mathematical imperfect iterative process, such as reciting the unending sequence
of positive integers.79 There are two particular reasons for mentioning the BMI here. First,
arguably one of the most accessible and compelling arguments that metaphor plays an ineliminable role in mathematics is that it seems to be required for an understanding of infinity,
among the most abstract and non-direct concepts in the human repertoire; it is thus perhaps
no surprise that extensive discussions of the BMI feature prominently in Where Mathematics
Comes From. Second, it is not clear what kind of conceptual metaphor the BMI is. It seems
findings suggest that the Yupnos may also be lacking portions of the ARITHMETIC IS MOTION ALONG A PATH and
ARITHMETIC IS USE OF A MEASURING STICK grounding metaphors. However, as Núñez does not even use the word
“metaphor” in the above article, it is difficult to interpret exactly how different he found the Yupnos’ suite
of conceptual metaphors to be from our own.
76
Lakoff and Núñez, Where Mathematics Comes From 150.
77
Lakoff and Núñez, Where Mathematics Comes From xii.
78
Lakoff and Núñez, Where Mathematics Comes From 160.
79
Lakoff and Núñez, Where Mathematics Comes From 165.
142
like a linking metaphor insofar as it is the connecting thread running between the conceptions of infinity arising in different mathematical domains.80 It also seems like a grounding
metaphor in that its ultimate source is the sensorimotor aspectual system.81 Pursuing this
point further, as noted above the general statement of the BMI is not inherently numerical
and is invoked non-mathematically; thus, a third possibility is that it is neither a grounding
nor a linking metaphor.82 Why this observation is noteworthy will become apparent in the
criticisms below.
For Lakoff and Núñez, “[m]athematics is a product of the neural capacities of our brains,
the nature of our bodies, our evolution, our environment, and our long social and cultural
history.”83 As such, embodied mathematics can be seen as a variety of constructivism, albeit one that is empirically based and descriptive rather than prescriptive. It is conceptual
metaphor that sets this theory apart from its traditional predecessors, binding the above
factors together and allowing mathematics to climb to ever loftier abstract heights while
tethering it to our most basic human capacities and experiences. It is this metaphorical
piggybacking of our rudimentary mathematical capacities upon our even more fundamental
sensorimotor systems that establishes embodied mathematics’ mathematics-as-practice orientation. Dehaene’s experimental findings support the Lakovian view: “we use brain circuits
to accomplish mathematical tasks that also serve to guide our hands and eyes in space —
circuits that are present in the monkey brain, and certainly did not evolve for mathematics,
but have been preempted and put to use in a different domain.”84 Mathematics is something
we have evolved to do to better function in the world.
The recognition that certain connexions in mathematics are metaphorical rather than
literal allows embodied mathematics to incorporate traditional insights while circumventing
many long-standing difficulties; I will return to discuss this aspect of the theory in the last
section of the chapter. However, as we saw in chapter 3, avoiding traditional problems
80
Lakoff and Núñez, Where Mathematics Comes From 161.
Lakoff and Núñez, Where Mathematics Comes From 156.
82
The BMI is clearly not an extraneous metaphor, as it cannot be eliminated without making a substantive
change to the conceptual structure of mathematics (Lakoff and Núñez, Where Mathematics Comes From 53).
83
Where Mathematics Comes From 9.
84
Dehaene 246. Dehaene uses the term neuronal recycling rather than referring to the piggybacking phenomenon as conceptual metaphor.
81
143
does not place CMT beyond scrutiny: the theory does not seem as firmly established as
Lakoff would have us believe. Likewise, embodied mathematics is a controversial theory,
and has been criticized by mathematicians and philosophers alike. Most published reviews
target Where Mathematics Comes From as a stand-alone work rather than as a fragment
of the Lakovian oeuvre upon which it thoroughly depends. This may contribute to the
fundamental misunderstandings behind many of the criticisms that cause them to fail. On
the other hand, some criticisms of Where Mathematics Comes From are consistent with those
directed at CMT generally despite their detachment from the greater corpus, suggesting that
these objections may be more substantive. Considering a selection of criticisms will help
expose the strengths and weaknesses of embodied mathematics, and suggest directions for
future development.
Embodied mathematics is subject to most of the frequently accurate yet nonfatal shallow
criticisms directed at CMT discussed in chapter 3 above. Lakoff and Núñez are not always
the best advocates for their own theory. Their writing can be nicely accessible, but is often
stylistically off-putting in various ways. At times, Lakoff and Núñez seem to hubristically
overstate the novelty of their mathematical interpretations.85 They have a tendency to
caricature rival positions, challenging straw men rather than the nuanced views of specific
authors; as Presmeg puts it:
I had the feeling that at times, in their zeal to destroy the myth [of the Romance
of Mathematics], they were tilting at windmills. . . For instance, in the section
titled ‘A Question of Faith: Does Mathematics Exist Outside Us?’ the authors
appeared to be setting up arguments solely for the purpose of tearing them down
(pp. 342-343). The resulting picture appeared to be a caricature, and I questioned
whether anyone in fact believed the Romance view in the extreme form in which
it was portrayed.86
The first printing of the book contains a number of acknowledged mathematical errors, though
these were corrected with published errata and in subsequent printings.87 In general, Where
Mathematics Comes From comes across as slipperier than most works in the philosophy of
85
Gerald A. Goldin, “Review: Counting on the Metaphorical,” Nature 413.6851 (2001): 19.
Norma Presmeg, “Review — Mathematical Idea Analysis: A Science of Embodied Mathematics,” Journal
for Research in Mathematics Education 33.1 (2002): 61.
87
George Lakoff and Rafael Núñez, “Reply to Bonnie Gold’s Review of Where Mathematics Comes From,”
MathDL, The Mathematical Association of America, 5 May 2009.
86
144
mathematics; while a certain degree of looseness is to be expected and tolerated given the
authors’ beliefs about metaphor and thought, even Lakoff and Núñez concede that “there
are indeed some passages in which we stated things in a sloppy way and we apologize.”88
There are many passages in the book where details are scant and elaboration or, at the very
least, specific references — particularly to other works in the CMT corpus and evidential
sources — would be most welcome. While none of these observations constitute a devastating
objection to embodied mathematics, they can affect readers with a dismissive predisposition,
preventing them from seeing past the presentational problems to give the underlying theory
a fair assessment.
Even if one is able to successfully exercise philosophical charity and avoid becoming biased against Lakoff and Núñez on account of their atypical style, the scantness of certain
key passages in the book certainly contributes to bringing about the misunderstandings that
plague many of the objections raised against embodied mathematics. In an uncharacteristic
move, Lakoff responds to a review of Where Mathematics Comes From by the mathematician
Bonnie Gold, observing that several of her comments indicate a fundamental misunderstanding of their position.89 In particular, they take Gold’s claim that their arguments in favor of
human mathematics “have little direct connection with the rest of the book” and the total
absence of the word “embodiment” from her review as evidence that she has fundamentally
missed the central importance of embodiment in their theory.90 By stripping the embodied
human element from their theory, Gold ends up misinterpreting their cognitive analyses as
mathematical ones and viewing conceptual metaphors as “essentially isomorphisms,” and,
accordingly, puts forth a variety of fundamentally flawed criticisms of the specifics of their
idea analysis.91 Though Lakoff and Núñez’s only published response targets Gold’s review
specifically, it is clear that other critics have also fundamentally misunderstood their position.
88
“Reply to Bonnie Gold’s Review.”
Lakoff and Núñez, “Reply to Bonnie Gold’s Review.”
90
Lakoff and Núñez, “Reply to Bonnie Gold’s Review.” Presmeg’s speculation that Lakoff and Núñez’s
discussion of neuroscience and infant arithmetic is ultimately distracting and unnecessary seems to betray a
similar misunderstanding on her part (Presmeg 60).
91
Bonnie Gold, “Read This! The MAA Online book review column: Where Mathematics Comes From: How
the Embodied Mind Brings Mathematics Into Being,” MathDL, The Mathematical Association of America,
5 May 2009. It should be noted that the conceptual metaphor CONCEPTUAL METAPHORS ARE ISOMORPHISMS may
be a fruitful one in a different context, when one is trying to understand conceptual metaphor in terms of
mathematics rather than vice versa; for more on such ideas, see chapter 5.
89
145
Whereas most criticisms of Lakoff and Núñez come from book reviews written by mathematicians, the philosophers Parsons and Brown are among the few to have published a
critical article responding to Where Mathematics Comes From. Their article contains a variety of criticisms motivated by Lakoff and Núñez’s rejection of mathematical platonism and
the “Romance of Mathematics.” Parsons and Brown interpret the anti-platonic argument in
the following way:
1. Human mathematical thought is based on conceptual metaphor (i.e., its
statements attribute physical properties to numbers and sets) (empirical
result).
2. Human mathematical thought is true metaphorically but not literally (from
1).
3. If there are Platonic entities (numbers and sets), then statements describing
them are literally true (assumption).
4. Therefore, it is not the case that human mathematical thoughts truly describe Platonic entities, even if they do exist (from 2,3).92
Their objection focuses on the middle two statements of the argument:
This argument is fallacious because it depends on the claim that metaphorical
statements cannot be used to express literal truths, which is patently false. . . Often
the only way we can express a (real, literal) truth is by using a metaphor, by saying something that, taken literally, does not mean quite what we want to get
across, but that we know will have the correct metaphorical meaning for our
listeners.93
While Parsons and Brown’s counterargument seems plausible at first glance, it is based upon
an equivocation. Within the first premise, they transition from an understanding of metaphor
as conceptual to a traditional linguistic understanding of metaphor. It is the parenthetical
portion of the first premise that introduces the error: while the statements of mathematics do
metaphorically attribute physical properties to numbers and sets, this is merely a symptom
of conceptual metaphor, not the whole of it. This misunderstanding demonstrates how easily
one’s own paradigm can creep in and subvert a genuine attempt to understand embodied
mathematics in its own paradigmatic milieu: Parsons and Brown clearly and explicitly note
92
Glenn G. Parsons and James Robert Brown, “Platonism, Metaphor, and Mathematics,” Dialogue 43.1
(2004): 52.
93
Parsons and Brown 52.
146
in the first section of their paper that “[c]onceptual metaphors. . . are not to be understood
as linguistic phenomena.”94
Given the equivocation in premise 1, it becomes more difficult to interpret the already
obscure notion of metaphorical truth introduced in premise 2. Throughout their paper, Parsons and Brown rely on a traditional interpretation which can be approximated thusly: a
metaphorical truth is an instance of non-literal language used to express a different literal
truth. In this interpretation, metaphorical truth is secondary, as it is parasitically dependent
on literal truth. It is not far from such a traditional interpretation to the once-popular idea
that metaphor constitutes an occasionally useful but primarily obfuscating deviant use of language that can be — and usually ought to be — carefully paraphrased as the equivalent literal
statement.95 The traditional interpretation is incompatible with the conceptual metaphor
interpretation: a metaphorical truth is a truth expressed in language reflecting concepts
structured by non-extraneous conceptual metaphors.96 For Lakoff and Núñez, our concepts
are not mere imperfect mirrors of objective reality. They do depend on our experiences of
the world and its regularities, but are ultimately active human constructions as opposed to
passive resemblances. All truths depend on our embodied concepts, and most — if not all
— concepts are partially structured by conceptual metaphor. Therefore, most truths are at
least partially metaphorical under the conceptual interpretation. This is clearly not the case
in the traditional interpretation, where the majority of truths must be literal. Lakoff and
Núñez’s point is that there is an irreconcilable disparity between the platonist understanding
of concept formation (where concepts approximate objective, human-independent structures)
and the embodied understanding of concept formation (where all concepts are thoroughly
human, arising from necessarily embodied experiences).97 Parsons and Brown’s equivocation
reveals a misunderstanding of the theory of conceptual metaphor that masks this disparity,
94
Parsons and Brown 49.
Lakoff and Johnson, Philosophy in the Flesh 119
96
As far as I am aware, Lakoff and Núñez never use the term “metaphorical truth” in their work, but I
believe the above interpretation is satisfactorily consistent with their view.
97
Lakoff and Núñez do concede that the idea that platonic entities exist is not itself inconsistent with embodied mathematics, though they argue that there are ontological reasons to believe their existence untenable;
regardless, given the impossibility of accessing such entities, positing their existence is entirely superfluous
(Where Mathematics Comes From 342).
95
147
causing their central counterargument to fail.98
As was noted in chapter 3, Lakoff and Johnson have consistently maintained that a position known at first as experientialism and later as embodied realism is an integral component
of their theory, claiming that “you cannot simply peel off a theory of conceptual metaphor
from its grounding in embodied meaning and thought. You cannot give an adequate account of conceptual metaphor and other imaginative structures of understanding without
recognizing some form of embodied realism.”99 A brief summary of the embodied realist
position, distilled from the entire Lakovian corpus, will augment the paltry account provided
in Where Mathematics Comes From and help overcome misunderstandings like those experienced by the critics cited above. For the embodied realist, the basis of all human cognition
is embodied experience. Whether you are seeing a tree with your eyes, hearing a friend call
your name with your ears, feeling a pain in your back, or imagining a unicorn in your head,
your human body makes an ineliminable contribution. Experiences are interactive gestalts:
they arise as unified wholes that are only later teased apart through interpretation. Importantly, embodied experiences are conceived as incorporating contributions from both the
world and the capacities and faculties of our bodies, though these aspects cannot be clearly
and fully disentangled from each other. This entanglement means there is no “ontological
chasm” between subject and object (or mind and body, if you prefer) to be explained away.100
Embodied realism is committed to the existence of a stable, mind-independent world that
exhibits regularities, an ultimate material basis for reality.101 However, there is no way for
us to transcend our embodiment and gain unmediated access to that structure: there is no
such thing as a disembodied experience. Conversely, every body is situated in some worldly
environment or other at all times. As Mark Johnson puts it, “our structured experience is an
organism-environment interaction in which both poles are altered and transformed through
an ongoing historical process.”102
One important consequence of embodied realism is that it “requires us to give up the
98
An extended version of this response to Parsons and Brown can be found in Postnikoff (2008).
Johnson and Lakoff, “Why cognitive linguistics requires embodied realism” 245.
100
Lakoff and Johnson, Philosophy in the Flesh 93–4.
101
Lakoff and Johnson, Philosophy in the Flesh 90, 110.
102
The Body in the Mind 207.
99
148
illusion that there exists a unique correct description of any situation.”103 A key example will
help clarify this claim. Lakoff and Johnson observe that humans exhibit two different levels
of understanding when it comes to colours: a phenomenological level and a neurobiological
level. At the phenomenological level, we experience colours as inherent properties of objects,
leading us to make assertions like “That emerald is green.” At the neurobiological level,
we understand colours as neural activations resulting from interactions between cells in our
retinas and reflected light of specific wavelengths. Thus, “[a]t the neural level, green is
a multiplace interactional property, while at the phenomenological level, green is a oneplace predicate characterizing a property that inheres in an object.”104 We take statements
originating from both of these levels to be valuable and true, and yet they often contradict
each other. Privileging one of these levels of understanding and subjugating the other does
not adequately resolve the conflict between them, as such a move either hampers our ability
to communicate or does injury to our notion of
TRUTH,
or both. Embodied realism bypasses
this dilemma by positing that the truth of a sentence is relative to understanding (rather
than some objective, human-independent state of the world) and thus acknowledges the
legitimacy of both levels of understanding without contradiction, a move not possible within
most traditional theories.105
It is the embodied realist dimension of the Lakovian theory that defends it against charges
of psychologism and relativism. Such objections are frequently raised against constructivist
and empiricist theories of mathematics, and some may worry that viewpoints that prioritize
mathematics-as-practice are vulnerable to these criticisms as well. In particular, the Lakovian posit that truth is relative to understanding seems to make embodied mathematics a
target for antirelativistic objections. The most noteworthy champion of antipsychologism is
Frege. One of the main theses of his Grundlagen der Arithmetik is “that mathematics and
logic are not part of psychology, and that the objects and laws of mathematics and logic
are not defined, illuminated, proven true, or explained by psychological observations and
results.”106 Frege contends that taking mathematics to be essentially psychological leads to
103
Lakoff and Johnson, Philosophy in the Flesh 109.
Lakoff and Johnson, Philosophy in the Flesh 105.
105
Lakoff and Johnson, Philosophy in the Flesh 106.
106
Martin Kusch, “Psychologism,” Stanford Encyclopedia of Philosophy, 14 Apr. 2012.
104
149
a variety of problematic results, chief among which is that it makes mathematical truth subjective and mind-dependent. This is antithetical to the certainty and universality that seem
to underlie our deepest mathematical understandings. While contemporary practice-oriented
views do tend to discuss the psychological states people possess when doing mathematics,
this does not necessarily entail they hold mathematics to be constitutively psychological; that
is, regardless of one’s philosophical disposition it is presumably now non-controversial that
people use their brains when doing mathematics. Even Frege acknowledges that mathematics
involves thinking and that studying such practices is permissible: “It may, of course, serve
some purpose to investigate the ideas and changes of ideas which occur during the course of
mathematical thinking.”107 That being said, many practice-oriented positions are unrepentantly psychologistic — including Lakoff and Núñez’s — and owe some sort of response to
Frege.
The following crude rendering of Frege’s antipsychologistic argument provides context for
Lakoff’s response:
If mathematics is constitutively psychological, it is subjective. If mathematics
is subjective, it is not objective. If it is not objective, mathematics is radically
relative. But this is an absurdity and an affront. Therefore, mathematics cannot
be psychological.
Lakoff claims that Frege’s argument contains a fatal error. It falsely concludes that because
the subjective belongs to the realm of the psychological, the realm of the psychological
is entirely subjective.108 This erroneous argument reflects the primitive understanding of
psychology that was available to Frege; it is almost certainly not simply an instance of one
of the foremost logicians of all time bungling categorical containment like a novice. To help
put Frege’s understanding of psychology in context, consider that his Grundlagen (1884) was
published over a decade before Freud’s first musings on the unconscious in 1895.109 When
coupled with an appropriate contemporary understanding of psychology, Lakoff’s observation
provides the most promising general line of defense against Fregean antipsychologism. Frege’s
107
Gottlob Frege, The Foundations of Mathematics, Trans. J.L. Austin, 2nd ed. (New York: Harper and
Brothers, 1953), xviii.4
108
Lakoff and Johnson, Philosophy in the Flesh 462.
109
David L. Smith, Freud’s Philosophy of the Unconscious (Dordrecht, Kluwer: 1999), 54.
150
argument depends on the false dichotomy between objectivism and radical relativism that is
antithetical to embodied realism; as Mark Johnson puts it:
we ought to reject the false dichotomy according to which there are two opposite
and incompatible options: (a) Either there must be absolute, fixed value-neutral
standards of rationality and knowledge, or else (b) we collapse into an “anything goes” relativism, in which there are no standards whatever, and there is no
possibility for criticism.110
Embodied mathematics espouses not a radical “anything goes” relativism, but a tempered
relativism constrained by ubiquitous commonalities in human biology and the Terran environment that provide a non-absolute yet solid basis for mathematics. Mathematics is neither
ad hoc and groundless, nor grounded in human-independent absolutes, but is stably grounded
internally by congruences and consistencies in human experience.
The detailed story that Lakoff and Núñez tell about the biological origins of mathematics
that fends off charges of relativism has led some critics to voice a very different concern.
Rather than objecting that embodied mathematics is problematically arbitrary, several authors worry that the Lakovian story is too specific. As Goldin puts it, “[Lakoff and Núñez]
often seem to assume, quite unjustifiably, that each mathematical construct can be understood in only one such way — the one they have discovered — and that they have found
the real metaphor from which the mathematics originates.”111 If Goldin is correct, then embodied mathematics would appear to be inconsistent with CMT, which allows for multiple
legitimate ways of understanding; moreover, this concern stands independent of whether one
takes the individual metaphors in the mathematical idea analysis to be convincing or spurious.112 However, there are good reasons to think that most objections of this type miss
the mark. One of the main goals of Where Mathematics Comes From is to account for the
perceived characteristic features of mathematics — universality, precision, stability, and so
110
The Body in the Mind 196.
Goldin 19.
112
Of course, many critics further argue that certain specific parts of the idea analysis are overly simplistic,
or outright erroneous; see, for example, Parsons and Brown’s criticism of the Lakovian analysis of set theory
(54–5). While I also find some parts of Lakoff and Núñez’s analysis unconvincing, I will not consider such
objections here as my primary concern is with defending the CMT approach in general rather than their
specific analyses. While Lakoff’s stylistic tone may suggest certainty and finality, embodied mathematics
must be open to ongoing revision and amendment if it is to be an empirically responsible theory; Lakoff and
Núñez even admit that their cognitive approach to mathematics is in its preliminary stages, suggesting that
they would admit there is room for improvement and development (Where Mathematics Comes From 11).
111
151
forth — within the framework of CMT and embodied realism and therefore without recourse
to human-independent mathematical entities. As such, the book focuses on describing the
ubiquitous conceptual metaphors that provide the skeleton of understanding that grounds
our collective mathematics rather than the cultural and individual idiosyncrasies that flavour
that core understanding. Thus, while Lakoff and Núñez do contend that there exists a collection of conceptual metaphors that feature in the understanding of every mathematics user,
they do not claim that our understanding is restricted to or exhausted by these metaphors.
In their concluding summary, they explicitly note:
Human conceptual systems are not monolithic. They allow alternative versions of
concepts and multiple metaphorical perspectives of many (though by no means
all!) important aspects of our lives. Mathematics is every bit as conceptually
rich as any other part of the human conceptual system. Moreover, mathematics
allows for alternative visions and versions of concepts. There is not one notion
of infinity but many, not one formal logic but tens of thousands, not one concept
of number but a rich variety of alternatives, not one set theory or geometry or
statistics but a wide range of them — all mathematics!113
It would seem that Lakoff and Núñez are only guilty of failing to adequately discuss individual
and cultural variations in mathematical metaphors, not of ruling them out completely.
While this failure thus does not constitute a fatal inconsistency, it remains a deficiency
in the presentation of the Lakovian theory of embodied mathematics that requires attention.
Even conceding that the main focus of Where Mathematics Comes From is (and should be)
to provide an explanation of the ubiquitous core of mathematical understanding that unfolds
within all of us, it is somewhat staggering that so very little of the book is allotted to reminding the reader that variations can and do occur between individuals’ understandings of
mathematics, let alone to discussing the details of some of those variations. This deficiency
seems to be at the root of a couple of telling criticisms which may provide guidance for clarification and emendation. For one, readers not immersed in CMT who are engaging with the
text as a stand-alone work may perceive embodied mathematics as having a problematically
naturalistic bias; Madden is one such critic:
In my opinion . . . a naturalistic approach should certainly not dismiss the way
mathematicians share definitions with one another, understand and criticize one
113
Lakoff and Núñez, Where Mathematics Comes From 379.
152
another’s reasoning, and use a precise, if artificial logical language to put their
ideas in writing so that those ideas can be judged by the world. Surely we
can acknowledge a role for intuition without ignoring the ways that logic and
conventional rigor support the kind of knowledge that mathematicians build and
share.114
First, it should be said that Lakoff and Núñez do acknowledge the power and rigor of the
formal approach, and find a place for it within the theory of embodied mathematics by way
of the Formal Reduction Metaphor — more on this below. And yet, there is certainly much
more to be said about the connexions between the more informal symbol, language, and diagram use that seems to constitute the majority of our mathematical practices and the mostly
unconscious naturalistic ground of our mathematical ideas.115 Seen in this way, Madden’s
comment is a version of the most significant unresolved general criticism of CMT found in
chapter 3: that Lakoff tends to inadequately discuss the relationship between the conceptual
and the linguistic, subjugating the latter as merely symptomatic of the former. As such, the
suggestions made in that chapter — that both semiotics and the work of Cornelia Müller
provide promising approaches to rectifying the Lakovian conceptual/naturalistic bias — also
hold here. One specific concern that needs attention is the thought that some conceptual
metaphors, mathematical or otherwise, might be better understood as arising through conscious and intentional action rather than unconscious naturalistic processes. In particular,
discussion of how CMT accommodates the speech act that Searle calls declaration seems like
it could be quite fruitful. Declarations allow language users to invoke their authorial power
to create genuinely novel linguistic or institutional facts; the mathematically necessary activities of naming, defining, and introducing new rules are all instances of declaration.116 Thus,
acts of declaration provide one likely non-naturalistic mechanism for conceptual metaphor
development.
A related criticism, raised by Presmeg, involves the way metaphors are classified in Where
Mathematics Comes From:
114
Madden 1186.
The symbol, language, and diagram use in most mathematics is informal only in comparison to Principia
Mathematica and other strict works of logic; even the most informal mathematics will usually still seem
achingly formal to the layman.
116
John Searle, “A taxonomy of illocutionary actions,” Expression and Meaning (Cambridge: Cambridge
UP, 1979), 16–20.
115
153
Lakoff and Nunez have undervalued what they call extraneous metaphors (p. 53).
There are metaphors, sometimes idiosyncratic, that individual mathematicians
or students may construct in their learning experiences, which are a powerful
part of sense-making in those experiences. . . individually constructed metaphors
are an issue that belongs with consideration of effective teaching and learning of
mathematics, and in this field they cannot be relegated to the role of an epiphenomenon.117
Lakoff and Núñez do not make it clear whether the tripartite distinction between grounding,
linking, and extraneous metaphors is to be understood as exhaustive or not; a careful reading of their descriptions of these classes suggests some mathematical metaphors may exist
that do not fit well into any of the three (recall the observation that the BMI, arguably the
most important metaphor in the book, presents classificational problems).118 Such metaphors
would possess a non-mathematical yet non-basic source domain and make at least a minimal
contribution to one’s understanding of mathematics; as this seems to be precisely the kind
of metaphor that Presmeg has in mind, I will refer to them as Presmegian metaphors for
convenience. If a distinct fourth class of mathematical metaphors does exist then Lakoff and
Núñez should have given it at least a brief mention. On the other hand, if the classification
scheme is taken to be exhaustive, then any Presmegian metaphors would have to be classified
as either grounding metaphors or extraneous metaphors.119 It is plausible that if an idiosyncratic metaphor with a non-mathematical source domain made a substantial contribution to
a person’s understanding that Lakoff and Núñez would classify it as a grounding metaphor
even if the source domain was abstract and the grounding therefore indirect; their explanation of grounding metaphors is vague and few examples other than the four fundamental
grounding metaphors of arithmetic are explicitly indicated in the text. However, if Presmeg
is correct that Lakoff and Núñez would classify such metaphors as extraneous, then we seem
forced to acknowledge that there are grades of extraneity, as Presmegian metaphors involve
understanding in a way that facile catachrestic conveniences like “step function” do not.
117
Presmeg 62.
It would be nice to present a good example of such a metaphor, but, as mentioned previously, the nature
of conceptual metaphor makes it difficult to arrive at such an example. While I might dream up any number
of possible mappings, the process of “dreaming up” would tend to produce extraneous metaphors, which
would fail to be examples.
119
They would clearly not be linking metaphors: not only are the necessary conditions on this class clearly
laid out (i.e., source and target domains both mathematical), but the majority of Where Mathematics Comes
From is devoted to examples of linking metaphors that confirm the definition (53).
118
154
Moreover, combining this insight with the idea of non-naturalistic conceptual metaphor development mechanisms raised in the previous paragraph suggests that it may be occasionally
possible for an initially shallow catachrestic metaphor to develop into something substantial,
robust, and non-extraneous. This possibility suggests that a more-dynamic understanding of
metaphor classification is warranted.120
The idea that Lakvoian taxonomies miss out important dimensions of metaphor occurs
elsewhere in the critical literature. Schiralli and Sinclair recognize two distinctions missed by
Lakoff and Núñez that could significantly clarify the theory of embodied mathematics. First,
Schiralli and Sinclair observe that “[Where Mathematics Comes From] does not differentiate
the term ‘mathematics,’ nor indicate whether metaphor might function differently depending
on whether one is learning, doing, or using mathematics.”121 That there may be significant
differences between these modes seems plausible considering that concepts seem to be more
rigid when being used and more plastic when being learned. In particular, Lakoff and Núñez
do not discuss metaphors with mathematical source domains and non-mathematical target
domains which may play an important role in mathematical modeling.122 If this distinction is found to be significant, Müller’s work on creating a robust diachronic understanding
of metaphor to replace the living/dead distinction could provide useful insights. Second,
Schiralli and Sinclair distinguish between conceptual mathematics (mathematics as a group
practice, or discipline) and ideational mathematics (how individuals represent the public
mathematical concepts to themselves).123 The idea that there may be idiosyncratic variation
in how individuals represent mathematical concepts is closely related to Presmeg’s criticism;
indeed, Schiralli and Sinclair claim that
Lakoff and Núñez provided metaphorical pathways to the concepts of [conceptual mathematics], through grounding and linking metaphors, but these are not
necessarily the same pathways that an individual will follow in creating his or
her conceptions in [ideational mathematics]. Lakoff and Núñez do not distinguish
[ideational mathematics] from [conceptual mathematics]. In fact, they imply that
120
It seems plausible that Müller could make a significant contribution here.
Martin Schiralli and Nathalie Sinclair, “A Constructive Response to ‘Where Mathematics Comes From’,”
Educational Studies in Mathematics 52.1 (2003): 81.
122
For further discussion and examples of such metaphors, see chapter 5.
123
Schiralli and Sinclair 81. Note that this strongly mirrors Müller’s distinction between the levels of system
and use (12–3).
121
155
ideational and conceptual mathematics will be isomorphic.124
To sum up, Schiralli and Sinclair “do not deny the power and pervasiveness of metaphor”
but do worry that the picture of metaphor presented in Where Mathematics Comes From
is problematically lacking in complexity, and emphasize that further empirical research is
necessary to resolve these speculations.125
Where Mathematics Comes From is an ambitious first attempt to explain mathematics
within the Lakovian framework of CMT and embodied realism. While it is clear that there
is much more work to be done, and that some of Lakoff and Núñez’s specific analyses are
spurious, the core insight that conceptual metaphor plays a constitutive role in mathematical reasoning and provides a mechanism by which mathematics can be grounded in basic
experiences and primitive biological capacities is both appealing and promising. A variety of criticisms have been proffered against the theory. Many objections fail because they
fundamentally misunderstand the Lakovian position and revert to a traditional linguistic conception of metaphor. The telling criticisms are not devastating to embodied mathematics but
do indicate important directions for necessary development. A great deal of concern could
be alleviated simply through improved scholarship, particularly making explicit, clear, and
frequent citations.126 Clearly situating embodied mathematics with respect to the empirical
studies underlying it, the earlier portions of the Lakovian corpus it depends upon, and other
leading theories of mathematical development and practice would fortify the theory significantly. While addressing the first two of these strengthens the foundations of the theory and
helps avoid misunderstanding, exploring the relationships between embodied mathematics
and other theories may help fill in some of the lacunae. One of the key aims of this chapter is
to perform some of this clarificatory work. Even if all the existing empirical evidence behind
the theory were exposed, it seems a considerable amount of additional scientific research
would be necessary to thoroughly substantiate the theorizing which has taken place; this
dissertation indicates a few noteworthy recent results in the field, but otherwise does not aim
124
Schiralli and Sinclair 84.
Schiralli and Sinclair 88–9.
126
While some may rightly argue that Where Mathematics Comes From and many other books in the
Lakovian oeuvre are written more for the layperson than the specialist, this neither invalidates nor appeases
the specialists’ desire for a scholastically rigorous version of the theory.
125
156
to make a contribution in this regard. Whether one is persuaded that embodied mathematics
shows promise or not, it is worth briefly considering some competing theories of metaphorical
mathematics.
4.3
Yablo’s Mathematical Figuralism
While Lakoff and Núñez have produced the most systematic and comprehensive theory of
mathematics-as-metaphorical thus far, they are not the only scholars to have explored such
ideas. Some of the critics mentioned above — notably Schiralli and Sinclair — endorse
the idea that metaphor plays a constitutive role in mathematics, even if they disagree with
the particulars of Lakoff and Núñez’s approach. It is noteworthy that researchers in mathematics, in philosophy, and in mathematics education have independently published work
supporting the idea that metaphor plays a more important role in mathematics than traditionally thought. Perhaps the most well known of these is George Pólya’s two-volume
Mathematics and Plausible Reasoning. In this work, Pólya distinguishes demonstrative from
plausible reasoning, claiming that though both play crucial and complementary roles in
mathematics, the importance of the latter has frequently been eclipsed by the former; that
is, students are taught to produce rigorous proofs and calculations more than creative conjectures. Mathematics and Plausible Reasoning aims to rectify the imbalance between the
two types of mathematical reasoning. Though Pólya never uses the word “metaphor” in that
work, analogical reasoning is considered as a key species of plausible reasoning.127 Another
mathematician who has considered the connexions between mathematics and metaphor is
Yuri Manin, whose “Mathematics as Metaphor” suggests that mathematics be considered as
a “specialized dialect of the natural language,” which is, as he understands it, fundamentally metaphorical.128 Mathematics-education professor David Pimm has been arguing that
“[metaphor and analogy] are as central to the expression of mathematical meaning as they
are to the expression of meaning in natural language” since the late 1970s.129 And philoso127
George Pólya, Induction and Analogy in Mathematics (Princeton: Princeton UP, 1954), 13.
Yuri Manin, “Mathematics as Metaphor,” Proceedings of the International Congress of Mathematicians,
August 21–29, 1990 (Tokyo: Springer-Verlag, 1991), 1666.
129
David Pimm, “Metaphor and Analogy in Mathematics,” For the Learning of Mathematics 1.3 (1981):
47. Pimm initially adopts a comparison view of metaphor, but shifts to a Lakovian approach in later work
128
157
pher and award-winning poet Jan Zwicky has written several works on the importance of
metaphorical thought, including “Mathematical Analogy and Metaphorical Insight” in which
she argues that mathematical and metaphorical insight are analogous in hopes of establishing
the legitimacy of the latter to those who still believe metaphor to be an eliminable linguistic
garnish.130 However, after Lakoff and Núñez, it is arguably Stephen Yablo who makes the
strongest philosophical case for metaphor being an integral part of mathematical practice.
In “Go Figure: A Path through Fictionalism,” Yablo sketches a theory of mathematics
which he refers to as figuralism.131 His position takes its basic inspiration from the fictionalist
insights of Hartry Field and others, but improves upon them by carefully integrating Kendall
Walton’s account of representation as make-believe, among other elements. “Go Figure”
begins with a puzzle: people routinely utter sentences whose truth seems to depend on the
existence of entities they do not believe in. At least some mathematical sentences are prime
examples of this predicament. Quine provides a list of three ways one might deal with such a
sentence: paraphrase the sentence so that the result is free of the problematic commitment,
stop uttering that sentence, or acknowledge the commitment to the questionable entities.132
Yablo claims that a fourth possibility lurks in Quine’s writing, though he did not explicitly
include it on the list: understand the sentence as advanced in a make-believe or fictional
spirit.133 This response to the puzzle — fictionalism — has the advantage of allowing us to
have our cake and eat it too: it legitimates the worrisome sentences rather than modifying
or eliminating them, and it does so without requiring a significant change in our ontological commitments. While fictionalism thus seems to be an appealing approach, it has its
own suite of difficulties to overcome. The most basic instrumentalist versions of fictionalism
(such as the one proffered by Field) posit that we make-believe certain sentences are true in
(“Mathematical Metaphor” 31).
130
Jan Zwicky, “Mathematical Analogy and Metaphorical Insight,” The Mathematical Intelligencer 29.3
(2006): 9. It is important to note that, unlike the others, Zwicky does not argue that mathematics is
fundamentally metaphorical. She even claims “No one is seriously going to maintain that mathematical
analogies and metaphors are essentially the same thing,” though she believes they are relevantly similar and
deeply connected (Zwicky, “Mathematical Analogy” 6).
131
Several of Yablo’s papers address related issues, particularly “Does Ontology Rest on a Mistake?” (1998),
“Abstract Objects: A Case Study” (2002), and “The Myth of the Seven” (2005). For clarity and simplicity,
I have opted to focus on the treatment provided by Yablo in “Go Figure.”
132
Yablo, “Go Figure” 177.
133
Yablo, “Go Figure” 179.
158
order to serve some larger purpose, such as simplifying our theory; however, such positions
stop short of providing an account of the mechanism of make-believing or the justificatory
details underlying the fiction.134 According to Yablo, this raises a variety of problems, including having no apparent way to distinguish correct and incorrect make-believe utterances.135
Metafictionalism attempts to overcome this flaw of instrumentalist fictionalism by positing
that by “making as if to assert [some utterance] S, one is really asserting that S is the right
kind of thing to make as if to assert” within the game being played.136 While this does provide a ground for distinguishing correct from incorrect make-believe utterances, it does not
account for the apparent necessity of mathematical statements. It also leaves a problematic
gap between mathematical fictions and the world that creates several puzzles surrounding the
application of mathematical concepts: in metafictionalism, the underlying truths are about
the game, not the world.137
The first step to overcoming these issues is to recognize the difference between a sentence relying on rules and it being about those rules. Object fictionalism contends that a
statement is fictional if an associated game of make-believe connects it to a real-world scenario which obtains (Yablo calls such a real-world scenario the real content of the associated
fictional statement). For example, consider an Arthurian game in which a certain stick is
designated as Excalibur. The statement “Darrell is wielding Excalibur” is fictional (“truein-the-world-of-make-believe”) if Darrell is actually holding the designated stick.138 Thus, in
object fictionalism, the correctness of fictional statements relies upon the rules of a game of
134
For details on Field’s instrumentalist fictionalism, see his Science Without Numbers (1980) and Realism,
Mathematics, and Modality (1989).
135
Yablo, “Go Figure” 179–80.
136
Yablo, “Go Figure” 181.
137
Yablo, “Go Figure” 181.
138
Walton, “Metaphor and Prop Oriented Make-Believe” 39. In Walton’s theory of make-believe, real-world
objects and events that fictions depend on are called props. He compellingly argues that there may be an
emphasis in games of make-believe which make them either content-oriented (where the prop is a means to
the end of the fiction, a device which grounds the game) or prop-oriented (where the aim of the fiction is to
facilitate efficient communication and/or improved understanding about the prop — for example, describing
Italy as a boot). Note also that the designation that makes an ordinary stick into a prop of Excalibur
requires an explicit declarative speech act in usual circumstances. It thus seems clear that theories of makebelieve have a fundamental interest in the speech act of declaration insofar as it is a crucial mechanism for
introducing new rules into a game. Hoffman provides a promising account of the role of speech acts in fiction
in her dissertation “Mathematics as Make-Believe” (1999, pages 76–84), but focuses on directives to the
exclusion of declarations.
159
make-believe, but is grounded in real-world circumstances; this development overcomes many
of the problems associated with the previously mentioned versions of fictionalism. However,
mathematical object fictionalism is itself susceptible to a powerful objection: it has no resources to handle sentences which enumerate collections of numbers. In object fictionalism,
the statement “The number of continents on Earth is 7” is fictional because there are seven
continents in the world but there is no number 7 in the world; that is, objects in the world
directly ground the fictional statement. However, the statement “the number of even prime
numbers is 1” is problematic for the object fictionalist because its real content would have
to be that there is one even prime number in the world, a claim which no fictionalist can
accept. Further, consider the statement “there are no numbers” — fictionalist philosophers
of mathematics are prone to making this assertion. The associated fictional statement would
be “the number of numbers is 0,” which is apparently self-refuting.139
To get around this problem, Yablo draws a parallel between these apparently problematic
statements and English statements of a similar form. Consider the following situation: Harry
is a manure salesman who is delivering a tarp-covered trailer full of bovine excrement to
Frank. However, Frank is erroneously convinced that Harry is lying to him and that there is
no manure under the tarp. Attempting to persuade Frank that no swindling is taking place,
Harry exclaims “This bullshit isn’t bullshit!” Read in a naive, context-independent way, this
sentence appears to be self-refuting. However, when the first “bullshit” is taken literally and
the second is taken figuratively, Harry’s statement is understood to be true.140 Thus, Yablo
suggests that, like the literal and figurative interpretations of the word “bullshit” in this
example, number-talk can function in different ways that make sentences like those found in
the previous paragraph unproblematic. Specifically, he claims that numbers can function as
either representational aids (in referring to quantities, for example) or as things-represented
(when talking about numbers), and that both functions sometimes occur within the same
sentence (when talking about quantities of numbers, for example). Further, Yablo claims
that such self-applied number-talk necessitates a mild relativism insofar as there are multiple
legitimate interpretations of sentences such as “there are not many even primes.” That is,
139
140
Yablo, “Go Figure” 184.
Yablo, “Go Figure” 185.
160
the fictionalist can engage with the parasitic number game in their interpretation of this
sentence when doing math (“there is only one even prime number, the rest are odd”) or
disengage from it when talking about math (“there are no even numbers, as numbers do not
exist”).141 Yablo refers to object fictionalism thus amended as figuralism to highlight the
strong parallels between figurative language use and number-talk; the ubiquity of figurative
utterances with multiple legitimate context-dependent interpretations is seen as vindicating
the relativism of his approach.142
Figuralism provides a systematic non-platonist answer to both its motivating puzzle and
a variety of related questions about mathematical language use.143 However, it is a controversial theory and has been criticized accordingly. The most noteworthy criticism, put
forward by Burgess and Rosen, is a variation on their general argument against mathematical
nominalism. The argument turns on the following distinction. The hermeneutic fictionalist
contends that “mathematicians’ own understanding of their talk of mathematical entities
is that it is a form of fiction, or akin to fiction: mathematics is like novels, fables, and so
on in being a body of falsehoods not intended to be taken for true.”144 The revolutionary
fictionalist maintains that mathematicians understand their utterances to be literally true
rather than fictional and are therefore systematically mistaken. Thus, hermeneutic fictionalists seek to provide a descriptive interpretation of mathematicians’ practices that reveals the
lack of an ontological commitment to abstract mathematical entities, whereas the aims of
revolutionary fictionalists are prescriptive, suggesting how to correct mathematics’ mistaken
dependence on said commitment. The essence of Burgess and Rosen’s argument is that
this distinction exhaustively partitions fictionalisms, and that both varieties are problem141
Yablo, “Go Figure” 185. The idea of a parasitic game becomes necessary when considering engaged selfapplied number-talk: whereas the fictionality of applied number-talk is directly grounded upon real-world
circumstances, self-applied number fictions are indirectly grounded and depend on circumstances within
applied number games for their correctness. In Waltonian terms, disengaged self-applied number talk is prop
oriented as the props in the parasitic game are parts of the basic number game (Yablo, “The Myth of the
Seven” 233–4). Note that “the appearance of stratification here is somewhat misleading, for parasitic games
tend to swallow their hosts; instead of two games, one parasitic on the other, we wind up with a single game
parasitic on itself” (Yablo, “Go Figure” 189). To me this seems analogous to how, in a sense, English serves
as its own metalanguage.
142
Yablo, “Go Figure” 191.
143
Yablo, “Go Figure” 195–6.
144
John P. Burgess, “Mathematics and Bleak House,” Philosophia Mathematica 12.1 (2004): 23; emphasis
his.
161
atic. Hermeneutic fictionalism fails because there is a distinct lack of empirical evidence that
mathematicians interpret their findings as fictitious. Revolutionary fictionalism fails because
there is overwhelming empirical evidence that mathematics works superbly as is, leaving us
wondering why the purported mistakes require correction if they have no apparent impact
on mathematics’ success.145
Given that Yablo says that figuralism was conceived in a “hermeneutic spirit,” one might
expect a defense specifically addressing the anti-hermeneutic horn of the argument.146 However, it seems that the best defense strategies instead claim that the hermeneutic/revolutionary
dilemma does not apply to figuralism in the way Burgess and Rosen suggest. One possible
approach would be to deny that the distinction is exhaustive and claim that figuralism fits
into some non-hermeneutic, non-revolutionary third category. This approach is potentially
promising, but Burgess argues against it and I will not discuss it further here.147 Sarah
Hoffman argues figuralism has both hermeneutic and revolutionary aspects, and that these
aspects are not problematic in the way Burgess and Rosen suggest. Yablo’s view is only
hermeneutic insofar as it is descriptive rather than prescriptive: he does not seek to alter
mathematical practice. How, then, can figuralism be revolutionary if it is not prescriptive?
The answer is that its prescriptions “do not seek to change mathematical practice or talk,
but rather to depose the philosophical view that we must believe that there are mathematical
objects in order to understand the way that (pure and applied) mathematics functions.”148
That is, Yablo’s figuralist revolution would take place within the philosophy of mathematics,
not within mathematics itself. Yablo’s line of defense takes a slightly different approach than
Hoffman’s, though the two seem to cohere with each other. Yablo explicitly rejects nominalism because he feels its negative ontological conclusions (i.e., that mathematical objects do
not exist) are stronger than warranted.149 Therefore, insofar as Burgess and Rosen’s argu145
Sarah Hoffman, “Yablo’s Figuralist Account of Mathematics,” 2012, Unpublished Article, 6.
Yablo, “Go Figure” 191.
147
Burgess 23n6.
148
Hoffman, “Yablo” 13.
149
Stephen Yablo, “Why I am Not a Nominalist,” Yablo’s personal website (2003). Yablo specifically requests
that this work not be quoted as it is only an incomplete draft, but I have chosen to reference it because
a) its title is very telling and b) it arguably provides the clearest presentation of certain claims that are
consistent with commitments he makes elsewhere in publication. For example, in “Does Ontology Rest on a
Mistake?” and “Must Existence-Questions Have Answers?” he contrasts both platonism and nominalism with
an ontological stance that he calls quizzicalism): “Quizzicalists. . . find it hard to take (some? all?) ontological
146
162
ment targets nominalist theories, Yablo’s figuralism seems immune to it. Though he denies
that he is a nominalist, Yablo claims that he is nonetheless a hermeneutic fictionalist, arguing
that the two viewpoints are distinct and that there is thus no inconsistency: hermeneutic
fictionalism holds that mathematical utterances do not carry a commitment to numbers,
while nominalism makes the stronger claim that numbers do not exist.150
The difference between Yablo’s and Burgess’ characterizations of hermeneutic fictionalism
points to a related criticism of figuralism that is more relevant to the present study: Burgess
claims that Yablo’s understanding of the literal/figurative distinction is problematic. In
particular, Burgess claims that it is unlikely that mathematicians are speaking figuratively
insofar as the non-literal use of language seems to require a conscious intention to diverge
from the default literal usage; he thus places the burden of proof on Yablo to empirically
confirm that such an intention indeed exists.151 Yablo counters not by providing proof that
mathematicians intend to speak figuratively, but instead with psychological evidence that
conscious recognition of non-literality is not a necessary component of successful figurative
exchanges.152 The understanding of figurative language grounding Burgess’ criticism seems
to belong to the traditional linguistic approaches that the first half of this dissertation argued
against, and thus Yablo’s defense might seem decisive. On the other hand, Yablo’s account
of metaphor could benefit from some clarification. That any amount of clarification would
be welcome is perhaps best illustrated by the following passage:
“say what you like about [the analytic/synthetic distinction], compared to the
literal/metaphorical distinction it is a marvel of philosophical clarity and precision. Even those with use for the notion admit that the boundaries of the literal
are about as blurry as they could be, the clear cases on either side enclosing a
vast interior region of indeterminacy.”153
What is clear is that Yablo’s approach is based primarily upon Walton’s theory of metaphor
as prop oriented make-believe, and that he holds — along with many contemporary metaphor
debate seriously and hold out little hope for a successful resolution” (Yablo, “Must Existence-Questions Have
Answers?” 297).
150
Yablo, “Why I am Not a Nominalist.”
151
Burgess 26.
152
Yablo cites Gibbs here: “Figurative language interpretation does not follow after an obligatory literal
misanalysis” (The Poetics of Mind 109).
153
Yablo, “Does Ontology Rest on a Mistake?” Things (New York: Oxford UP, 2010), 120.
163
theorists — that figurative language is neither rare nor exceptional; this provides a promising
foundation for clarificatory efforts. One of the key results of the final section of this chapter is
that CMT and embodied mathematics can help clarify and strengthen Yablonian figuralism,
and vice versa.
4.4
Conceptual Metaphor Theory and the Philosophy
of Mathematics
This chapter has surveyed a variety of mathematical viewpoints, each possessing distinctive
strengths and weaknesses. All of the traditional viewpoints considered at the beginning of
the chapter were ultimately rejected, and yet each seems to capture some important aspect
of mathematics that traditional rival theories have difficulty accommodating. Unfortunately,
the traditional viewpoints have usually been seen as strongly incompatible with one another,
occasionally with tragic results.154 Theories of mathematics typically do not exist in isolation
but form part of a larger system of thought; these incompatibilities often derive from (or
entail) more fundamental disagreements about the nature of truth or reality. Incompatible
theories of truth traditionally leave little to no room for reconciliation within some overarching
theory, largely due to their strictly antirelativist commitments. One of the primary strengths
of CMT is that because it is a general theory of concepts it functions as both a philosophy
of mathematics as well as a philosophy of philosophies of mathematics.155 In this latter role,
CMT is promisingly inclusionary: conceptual metaphor provides a mechanism for integrating
the strongest aspects of various inconsistent theories of mathematics into a coherent whole.
This final section of this chapter sketches compatibilities between embodied mathematics
and other mathematical theories, strengthening the explanatory power of the theory and
154
Kronecker’s intuitionist objections to Cantor’s transfinite arguments infamously generated professional
conflict that prevented Cantor from obtaining a professorship worthy of his talent, and ultimately contributed
to the mental health issues that plagued him in his later years (Barrow 198–204). A similar (though somewhat
less tragic) public rivalry between the intuitionist Brouwer and the formalist Hilbert was dubbed the FroschMäusekrieg (or War of the Frogs and the Mice) by Einstein (Barrow 216–26).
155
Part III of Lakoff and Johnson’s Philosophy in the Flesh considers various major philosophical movements
throughout history from the CMT viewpoint, describing the predominant conceptual metaphors underlying
each.
164
suggesting directions for future research and development.
The claim that CMT can form the basis of an integrated philosophy of mathematics
may seem immediately false given that Lakoff and Núñez say that “Mind-based mathematics,
as we describe it in this book, is not consistent with any of the existing philosophies of
mathematics: Platonism, intuitionism, and formalism. Nor is it consistent with recent postmodernist accounts of mathematics as a purely social construction.”156 Additionally, Lakoff
and Núñez take particular issue with the platonistic “Romance of Mathematics,” arguing
that conceiving of mathematics as transcendental rather than constitutively embodied is
not merely empirically unsubstantiated but harmless, but rather that dehumanizing and
elevating mathematics does social harm by making the subject seem arcane and frustratingly
inaccessible to many students.157 Despite Lakoff and Núñez’s vehement opposition to fullfledged platonism, embodied mathematics allows for — and, indeed, even requires — certain
aspects of mathematical platonism; if embodied mathematics can successfully incorporate
elements of its staunchest rival theory, it is plausible that it can integrate aspects of other
theories as well.
It is worth noting that Lakoff and Núñez do not definitively reject the possibility of the
existence of a transcendental mathematics. What they do argue is that there is no (and can be
no) evidence establishing the existence of transcendental mathematical objects, and that even
if they were to exist, our embodied mathematical concepts would be independent from them,
not representations of them.158 This kind of compatibility is so minimal it is hardly worth
mentioning, but there is a more robust sense in which platonism and embodied mathematics
are compatible. Lakoff and Núñez acknowledge that we do, in part, conceptualize numbers
as objects, and have no desire to eliminate such talk. They claim that the four grounding
metaphors of arithmetic together naturally induce the more general conceptual metaphor
NUMBERS ARE THINGS IN THE WORLD.
159
Their brief discussion asserts that this metaphor is
an integral part of our understanding of mathematics, allowing for efficient communication
and providing structure for important notions such as mathematical closure by way of its
156
Where Mathematics Comes From 9; emphasis theirs.
Where Mathematics Comes From 341.
158
Lakoff and Núñez, Where Mathematics Comes From 4.
159
Lakoff and Núñez, Where Mathematics Comes From 80.
157
165
inference-preserving nature.160 While Lakoff and Núñez see NUMBERS ARE THINGS IN THE WORLD
as a natural and useful part of mathematics, at the level of the philosophy of mathematics,
they wish to acknowledge its metaphoricity and therefore disallow its use in making stronger
metaphysical inferences. In this, there seem to be important similarities between the Lakovian
and Yablonian approaches to mathematical object talk, connexions that will be discussed
shortly.
As was the case with platonism, embodied mathematics embraces the mathematical practices associated with formalism while simultaneously acknowledging its metaphoricity and
rejecting its philosophical pretensions.161 That is, Lakoff and Núñez endorse the idea that
formal symbolization and the axiomatic method are an important aspect of mathematics, but
reject the idea that mathematics is reducible to such an approach: “Mathematics is not literally reducible to set theory in a way that preserves conceptual differences. However, ingenious
metaphors linking ordered pairs to sets and numbers to sets have been explicitly constructed
and give rise to interesting mathematics.”162 They claim that there is a metaphorical schema
— dubbed the
FORMAL REDUCTION METAPHOR
— which allows all mathematical notions to be
conceptualized in terms of axiomatic set theory: “the Formal Reduction Metaphor inherently
makes an interesting claim: The relatively impoverished conceptual structure of set theory
and formal logic is sufficient to characterize the structure (though not the cognitive content)
of every mathematical proof in every branch of mathematics.”163 The
METAPHOR
FORMAL REDUCTION
is a collection of linking metaphors that provide unifying structure to mathemat-
ics, connecting disparate branches through a set theoretic hub.164 Traditional formalists see
160
Lakoff and Núñez, Where Mathematics Comes From 81. Performing a mathematical operation on elements of a given set sometimes yields a result that is not contained in the set, such as when one subtracts
a larger natural number from a smaller one. Informally, the idea of closure is that such a set can often be
enriched with further elements so that the result of performing the operation on the elements of the enriched
set always lands within the enriched set. Pimm (1981) also discusses the role of metaphor in closure.
161
As was discussed at the beginning of the chapter, the boundary between logicism and formalism becomes
blurred when one focuses on mathematical practice without considering the philosophical underpinnings. It
seems likely that this is one of the main reasons Lakoff and Núñez only use the word “logicism” once, in
passing, in Where Mathematics Comes From. Following their lead, only formalism will be discussed in this
paragraph, but the reader should understand the discussion to pertain to logicism as well.
162
Lakoff and Núñez, Where Mathematics Comes From 152.
163
Lakoff and Núñez, Where Mathematics Comes From 373.
164
Whether it constitutes part of the FORMAL REDUCTION METAPHOR or is merely associated with it, most
explanations of formalism invoke the metaphor MATHEMATICS IS A GAME.
166
these connexions as presenting an opportunity to reduce mathematics to set theory, thereby
avoiding multiplication of entities. Lakoff and Núñez argue that it is desirable to view these
connexions as allowing for increased richness of structure in mathematics through conceptual blending rather than to use them to eliminate other branches of mathematics through
reduction, which would lead to a starker, more restrictive mathematics; part of the evidence
that they present in favour of their interpretation is that the experimental data conflicts
with reductivist formalism. Additionally, formalistic reduction has difficulties taking into
account the various versions of formal logic and set theory.165 There are more dimensions to
our mathematical conceptualizations than set theory alone affords, and the fascinating and
powerful capacity of mathematics to cross-model itself should be embraced and utilized, not
invoked self-defeatingly.166
While embodied mathematics incorporates the mathematical practices of the platonist
and the formalist while rejecting their philosophical claims, the opposite holds true in its
relationship with most constructivists. Like constructivism, embodied realism posits that
mathematics is a construct of the human mind: “If there are ‘foundations’ for mathematics,
they are conceptual foundations — mind-based foundations.”167 Lakoff and Núñez’s account
of how limited inborn numerical capacities such as subitization give rise to the full panoply
of quantitative mathematics brings to mind Kronecker’s famous dictum: “The integers were
made by God; all else is the work of man.”168 However, embodied mathematics rejects the
intuitionist restriction of mathematical practice to constructive methods as unwarranted and
needlessly limiting. Lakoff and Núñez claim that the law of the excluded middle — generally
disallowed to some extent in intuitionist mathematics — emerges naturally in our understanding of categories, transfered via conceptual metaphor from the spatial logic inherent in
the container schema.169 Whereas intuitionists do not admit infinity into their mathemati-
165
Where Mathematics Comes From 374.
Proponents of embodied mathematics hold that logic depends on the embodied mind, and thus there
is a neurological conceptual system behind all axiomatics. Applying Occam’s razor from within the logical
domain to cut off all non-set-theoretic aspects of our mathematical concepts seems to constitute a kind of
category mistake, unless one assumes that all reasoning and conceptualization is axiomatic logic all the way
down. However, this begs the question.
167
Lakoff and Núñez, Where Mathematics Comes From 376; emphasis theirs.
168
Barrow 188.
169
Where Mathematics Comes From 44.
166
167
cal practices (as finite humans cannot literally construct infinite objects), using the BMI to
explain how humans metaphorically create infinities is one of the main objectives of Where
Mathematics Comes From.170 Thus, while Lakoff and Núñez and the intuitionists agree that
mathematics is a human construct, they differ significantly on what is being constructed and
how.171
Embodied mathematics is consistent with weak social constructivism insofar as it acknowledges that cultural factors and historical accidents play an important role in the development
of mathematics:
Many of the most important ideas in mathematics have come not out of mathematics itself, but arise from more general aspects of culture. The reason is obvious.
Mathematics always occurs in a cultural setting. General cultural worldviews will
naturally apply to mathematics as a special case. In some cases, the result will
be a major change in the content of mathematics itself.172
However, unlike more radical social constructivist theories, embodied mathematics “explicitly rejects any possible claim that mathematics is arbitrarily shaped by history and culture
alone.”173 Radical social constructivism tends to be associated with problematic “anything
goes” relativism. Accordingly, such theories have inadequate resources to account for the
perceived nonarbitrariness of mathematics. Lakoff and Núñez, on the other hand, hold that
culture and history are merely two among many factors contributing to mathematical development, and it is embodiment that prevents mathematics from being arbitrary. As they put
it,
Where does mathematics come from? It comes from us! We create it, but it
is not arbitrary—not a mere historically contingent social construction. What
makes mathematics nonarbitrary is that it uses the basic conceptual mechanisms
of the embodied human mind as it has evolved in the real world. Mathematics
is a product of the neural capacities of our brains, the nature of our bodies, our
evolution, our environment, and our long social and cultural history.174
170
I realize that there are various strains of intuitionism, and many of them are not strictly finitistic; however,
even those intuitionisms that do admit some idea of infinity will have a limited understanding compared to
the non-intuitionist.
171
Though Lakoff and Núñez explicitly tell us that embodied mathematics is inconsistent with intuitionism,
this is the only time they use the word in their book; they never elaborate on the inconsistency (Where
Mathematics Comes From 9).
172
Lakoff and Núñez, Where Mathematics Comes From 358.
173
Lakoff and Núñez, Where Mathematics Comes From 362; emphasis theirs.
174
Lakoff and Núñez, Where Mathematics Comes From 9.
168
Cultural and historical factors may be of peripheral importance compared to embodiment in
the Lakovian theory, but it is worth looking a little closer at the contribution they make.
For Lakoff and Núñez, embodied experience accounts for the stability, precision, consistency, discoverability, and universality of mathematics.175 Thus, if culture has a distinctive
role to play, it will involve the non-universal aspects of mathematical concepts. This is consistent with CMT in general, as per the explanation in chapter 3: while primary metaphors
tend to be ubiquitous due to their direct grounding, there is significantly more room for
cultural variation in the complex metaphors that structure our abstract concepts. A considerable amount of anthropological and sociological research has sought to find and explain
noteworthy conceptual differences between the mathematics of isolated peoples and our own;
recall, for example, the finding that the Yupnos of Papua New Guinea lack the linking
metaphor NUMBERS ARE POINTS ON A LINE. The relationship between culture and concept seems
complicated and bidirectional: not only do different cultures develop different concepts due
to different shared experiences, but some cultures may originate from the shared conceptual
commitments of their members. This seems particularly relevant in mathematics, where
a formalist may choose which axioms to adopt: a community of mathematicians who opt
to practice an alternative form of mathematics (non-Euclidean geometry, non-well-founded
set theory, etc.) may constitute a culture. Thus, for Lakoff and Núñez the very possibility
of inconsistent but equally legitimate coexisting alternative forms of mathematics exposes
mathematics’ cultural dimension.
Of course, such mathematical differences can occur vertically as well as laterally; that
is, mathematics exhibits both diachronic and synchronic variation. The apparently timeless
nature of mathematics has a tendency to obfuscate the history of mathematical concepts even
more than happens in other disciplines. If, for some reason, one does think about the history
of mathematics, thanks to the traditional dominance of platonism and formalism there is a
strong temptation to think of it as a gradual expansion of knowledge, analogous to either
the discovery and conquest of new, uninhabited territories or the meticulous piece-by-piece
assembly of a structure unfolding from a firm foundation into unoccupied space, or some
blend of the two. A key point of disanalogy between these two viewpoints and the history
175
Where Mathematics Comes From 350.
169
of mathematics comes in with the words “uninhabited” and “unoccupied”: mathematical
progress does not occur in a vacuum but rather is constrained and guided by accidental
elements such as puzzles and problems arising in other areas, technological advances, political
climate, etc.176 These constraints are often insufficient to uniquely determine the expansion
of mathematical knowledge, forcing mathematicians to make creative choices that influence
the way the discipline progresses. To illustrate, Lakoff and Núñez briefly discuss some ways
that the advent of the computer helped shape mathematics. On the one hand, computer
technology eventually facilitated computations and visualizations that were not previously
feasible, allowing for the development of fractal geometry, for example. On the other hand,
the digital nature of computer technology is not immediately well suited to mathematics
involving the continuum of real numbers. To improve the way computers approximate real
numbers and enhance the applicability of the technology, mathematicians chose to develop
new floating-point arithmetics. These arithmetics possess some features that would seem
very unusual to most people — such as considering positive zero and negative zero to be
distinct numbers — and yet, given the ubiquity of computers and the incredible rapidity
of their computations, the vast majority of arithmetic takes place within a floating-point
context.177
While Lakoff and Núñez acknowledge the historical dimension of mathematics, their mathematical idea analysis is predominantly synchronic; incorporating a more-thorough account
of the diachronics of mathematics is desirable. Establishing coherences between the Lakovian
theory and Imre Lakatos’ quasi-empiricist account of mathematics would go a considerable
distance towards attaining a balanced understanding of mathematical conceptual development. It is not possible to fully integrate the two theories here, but a groundwork for
synthesis can be sketched. In Proofs and Refutations — Lakatos’ primary work in the phi-
176
Even taboos can have a significant impact on the way mathematics unfolds. Two important examples
spring immediately to mind. The Pythagoreans believed that the entire universe could be explained in terms
of ratios of whole numbers, a commitment that made thought about irrational quantities blasphemous; those
who questioned the ultimate status of √
the rational numbers were allegedly exiled or put to death. Eventually,
demonstration of the irrationality of 2 was instrumental in the dissolution of the cult (Kline 32–3). The
Muslim proscription against idolatry meant that abstract geometric designs were often employed in decorating
their architecture, facilitating a productive, bidirectional connexion between decorative art and mathematical
theory (Islamic Art and Geometry 10).
177
Lakoff and Núñez, Where Mathematics Comes From, 360–1.
170
losophy of mathematics — he presents a picture of mathematical progress that is far more
complicated and tumultuous than the slick definitions and proofs that are the end result of
that process suggest. The majority of the book presents a beautiful case study in dialogue
format: a teacher and a classroom full of pupils discuss how to go about proving Euler’s
Polyhedron Theorem (the formulation which serves as the starting point of the dialogue: for
all polyhedra, Vertices + Faces - Edges = 2).178 This result is easily confirmed for specific
polyhedra — such as the Platonic and Archimedean solids — but such confirmations do not
constitute a proof, and thus the formula is not yet a theorem but only an inductively motivated conjecture.179 Lakatos’ dialogue follows the historical record, glossing many decades
of mathematical debate. A proof (Cauchy’s, in this particular case) is offered, lending substance to the formula but also providing opportunity and inspiration for the generation of
counterexamples based upon the specific assumptions invoked and methods utilized. Any
such counterexamples which arise invite a response that tends to fall somewhere between the
extremes of repudiation and capitulation, some qualification or clarification of the concepts
and techniques of the proof that takes the counter out of the examples. The improvements
made to the proof may provoke further counterexamples, resulting in a cyclic process of
conceptual negotiation and refinement. The details of this process of negotiation are the
principal focus of Lakatos’ book; unfortunately, a comprehensive discussion of his insights is
not possible here. Lakatos’ understanding of mathematical development is a blend of Popperian falsificationism and Hegelian thesis-antithesis-synthesis dialectic; mathematics, then,
is quasi-empirical insofar as it is a practice that proceeds like the physical sciences, only with
thought-experiments in place of laboratory work.180 Well-defined mathematical concepts
only emerge as the process of proofs and refutations reaches an equilibrium; thus, the often
“artificial and mystifyingly complicated” looking axioms and definitions are the end result of
mathematical practice and not the ex nihilo origin, as the clean axiomatic deductions that
178
Imre Lakatos, Proofs and Refutations (New York: Cambridge UP, 1976), 7.
The Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron) are familiar to most,
certainly to those who have ever played a role-playing game like Dungeons and Dragons requiring a variety
of non-standard dice (Sutton 2). The Archimedean solids are less commonly known: they are semi-regular
convex polyhedra with two (or more) distinct types of regular polygonal faces. There are 13 of them (15 if
chiral pairs are distinguished) (Sutton 32, 44).
180
Lakatos 9, 144–5, 149.
179
171
pave over the dynamic spadework suggest.181 Like Lakoff and Núñez, Lakatos warns against
attempting to ontologically reduce mathematics to the formal.182
The suggestion that mutually beneficial coherences exist between Lakoff and Lakatos is
not unprecedented: in Women, Fire, and Dangerous Things, Lakoff cites Lakatos in his short
list of noteworthy scholars involved in overturning the classical theory of concepts.183 Both
theories hold that concepts are not necessarily essentially definitional, and emphasize nonformalistic mathematical practice as fundamental. What Lakatos offers is a general account of
the dynamics of mathematical conceptual development, invoking the metaphor MATHEMATICAL
HISTORY IS A DIALOGUE
in the telling. Lakatos’ account of the concept
POLYHEDRON
seems
compatible with Lakoff’s theory of categorization: while certain polyhedra (such as cubes and
the other Platonic solids) are prototypical of the class, the counterexamples which arise force
mathematicians to make precise the radial structure of the category; thus understood, the
structure of POLYHEDRON and MOTHER appear similar, though the histories of the two concepts
are surely radically different.184 Thus, Lakatos provides an account of the development
of mathematical concepts drawn from historical evidence that is coherent with CMT and
therefore may augment the diachronic dimension of that theory. On the other hand, a
Lakatosian could use this coherence with Lakovian embodied realism to help counter charges
of relativism. Exploring the depth of this coherence and the fruitful syntheses it engenders
is a worthy task. This discussion must wait for another time, as there are other important
coherences to consider.
There are obvious connexions between metaphor and fiction. Both “metaphorical” and
“fictional” are used as antonyms for “literal,” a connexion backed up by the related common belief that, while accurate literal statements are true, fictions and metaphors are falsehoods.185 Though skillful critical thinking practice cautions against relying on negative definitions, this shared opposition indicates the two ideas are related to some degree. It is
181
Lakatos 142.
David Bloor, Knowledge and Social Imagery, 2nd ed. (Chicago: U of Chicago P, 1991), 152.
183
Women, Fire, and Dangerous Things 17.
184
Lakoff, Women, Fire, and Dangerous Things 83.
185
Recall that conceptual metaphor theorists reject the literal/metaphorical opposition, at least to the extent
that statements understood as obviously and directly true frequently invoke conceptual metaphors (Lakoff
and Johnson, Metaphors We Live By 5).
182
172
also clear that fiction and metaphor can and do interact with each other. Metaphors frequently occur in works of literary fiction: “What light through yonder window breaks? / It
is the east, and Juliet is the sun!”, one of the most frequently cited examples of linguistic
metaphor, is invoked by Shakespeare to convey fictional Romeo’s feelings and intentions towards fictional Juliet to the reader.186 Conversely, fictional entities and situations can figure
in linguistic metaphors, as in Searle’s example “Sally is a dragon.”187 Further, an extended
passage or even an entire fictional work can be read allegorically, used as a metaphorical
device for better understanding something apart from the fiction, regardless of the author’s
intentions. Finally, some philosophical work has been done towards establishing deeper connexions between metaphor and fiction, typically focused on trying to explain the former in
terms of the latter. For example, Kendall Walton argues that it is advantageous to understand at least some metaphors in terms of prop oriented make-believe.188 Many of Walton’s
writings are dedicated to developing a theory of fiction rooted in imagination, pretense, and
games of make-believe. Games of make-believe typically rely on props, “generators of fictional truths” which provide structure to the fiction; props considered by Walton range from
simple children’s toys to literary fiction and other representational artworks.189 The majority of Walton’s writing focuses on content oriented make-believe, where the focus is on the
fictional world of the game and associated props serve as a means to that end. However,
he does also give some attention to discussing prop oriented make-believe, where “interest is
focussed on the props themselves; the envisioned make-believe provides a way of describing
them.”190 Stage props featured in a theatrical production are an example of the former, while
referring to a person wearing spectacles as “four eyes” is an instance of the latter. Walton
claims that many metaphorical statements invoke a game of prop oriented make-believe; for
example, when a plumber refers to a certain fixture as “male” it is likely that she is doing
so merely as a referential convenience and that her focus is on the protruding (as opposed to
recessed) nature of the fixture rather than exploring a game where inanimate plumbing parts
186
William Shakespeare, Romeo and Juliet II.ii.
“Metaphor” 87.
188
“Metaphor and Prop Oriented Make-Believe” 45.
189
Kendall Walton, Mimesis as Make-Believe (Cambridge: Harvard UP, 1990), 37–8.
190
Walton, “Metaphor and Prop Oriented Make-Believe” 43.
187
173
are gendered.191 On the other hand, more elaborate, open-ended metaphors may “function
something like the stipulative launching of a (content oriented) game of make-believe, which
then grows naturally beyond the original stipulation.”192 Some metaphors may transition
between the two orientations; for example, this would be the case if the aforementioned
plumber began extending and enriching her fixture metaphor beyond male/female to include
hermaphrodites, eunuchs, the transgendered, and so forth. Though he is the first to admit
that he is not proposing a theory of metaphor, Walton’s observations provide an interesting
and useful way of thinking about metaphors.193
Considering these connexions, it is somewhat surprising that nobody has looked at integrating Lakovian embodied mathematics with mathematical fictionalism before now. All
versions of fictionalism draw an analogy between mathematics and fiction to help reconcile
our mathematical practices with a commitment to the non-existence of mathematical entities.194 In terms of CMT, mathematical fictionalism fundamentally depends on the metaphor
MATHEMATICS IS FICTION.
Interestingly, unlike platonists, most fictionalists acknowledge that a
metaphor underlies their viewpoint and that there are important disanalogies between mathematics and fiction. For people suspicious about the existence of abstract entities, there
seems to be a natural progression from using the platonist metaphor
THE WORLD
NUMBERS ARE THINGS IN
while doing mathematics, to recognizing it as a metaphor when thinking about
mathematics, to turning to the metaphor
MATHEMATICS IS FICTION
to help in making sense of
the metaphoricity of NUMBERS ARE THINGS IN THE WORLD; that is, recognizing that numbers are
not things in the world like frogs and toasters raises a suite of questions about what is going
on when sentences seem to refer to numbers, questions that
MATHEMATICS IS FICTION
can help
define. Thus, mathematical fictionalism can be understood in terms of CMT, and can enhance embodied mathematics by providing resources to help understand how mathematical
191
“Metaphor and Prop Oriented Make-Believe 40.
Walton, “Metaphor and Prop Oriented Make-Believe 53.
193
Walton, “Metaphor and Prop Oriented Make-Believe” 45. It is worth noting that Elisabeth Camp
argues that “metaphors generally involve a different sort of imaginative game than make-believe [“seeing-as”],
and that utterances which do evoke make believe are not usually metaphors” (“Two Varieties of Literary
Imagination” 110). Her arguments that pretense and seeing-as are distinct, important activities of the
imagination are compelling and certainly warrant further consideration. However, as my viewpoint does not
depend on reducing metaphor to fiction or fiction to metaphor, I will not discuss her position further here.
194
Mark Balaguer, “Fictionalism in the Philosophy of Mathematics,” Stanford Encyclopedia of Philosophy,
30 Jul. 2012.
192
174
sentences refer and what they are referring to.
Building on the above, the connexions and coherences between Lakovian embodied mathematics and Yablonian figuralism suggest that a successful partnership might be forged between the two theories. Figuralism differs from generic fictionalism in a few important ways
that makes it a better fit for embodied mathematics; these differences lead some to question
whether Yablo’s theory should be classified as a variety of fictionalism.195 Most fictionalists
assert that mathematical statements are false because they involve failed direct reference to
non-existent mathematical objects. Yablo says that mathematical statements can be true
because they may refer indirectly to an underlying real state of affairs by way of a game
of make-believe: “one gives voice to the real truth by making as if to assert the fictional
truth that it enables.”196 Thus, at the heart of Yablo’s view resides Walton’s prop oriented
make-believe applied to the case of mathematical utterances. Walton, then, is one connexion
between Yablo and Lakoff: in “Metaphor and Prop Oriented Make-Believe,” he cites Lakoff
and Johnson and uses ARGUMENT IS WAR as an explicit example of prop oriented make-believe,
providing a brief analysis that seems to cohere with CMT.197 This is not the only connexion between the two theories. Both viewpoints are mildly relativist: Yablo presents the term
relative reflexive fictionalism as practically synonymous with figuralism, and notes that distinguishing engaged and disengaged modes of understanding involves “a kind of relativism”.198
Both viewpoints are practice-oriented, taking the empirical observation that our mathematical practices are successful as fundamental. Indeed, both Lakoff and Yablo concede that,
despite doubts about the existence of mathematical objects, it is possible that they could
have an objective existence beyond our ken; however, precisely because such objects would be
beyond our experience, their ultimate ontological status must be irrelevant to mathematics
as we know it.199 Yablo’s description of object fictionalism’s Waltonian core makes heavy use
of the metaphor
FICTIONAL STATEMENTS ARE EXPONENTIAL FUNCTIONS,
a move that conceptual
metaphor theorists can both understand and appreciate.200 And, of course, last but not least,
195
Balaguer, “Fictionalism in the Philsophy of Mathematics.”
“Go Figure” 183.
197
“Metaphor and Prop Oriented Make-Believe” 45.
198
“Go Figure” 189.
199
Yablo, “Go Figure” 193; Lakoff and Núñez, Where Mathematics Comes From 342–3.
200
“Go Figure” 182.
196
175
both theories understand mathematical practice as fundamentally involving metaphor.
So the two theories seem to have enough in common to play well together. What motivation is there for attempting to integrate them? That is, what can Lakoff and Yablo provide
each other? Despite the various commonalities discussed above, figuralism and embodied
mathematics come at explaining mathematical practice from different directions. Embodied
mathematics works from the inside out, sketching how our vast system of mathematical concepts emerges from basic biological capacities and common experiences. Figuralism works
from the outside in, focusing on how mathematical language functions. Thus, the two viewpoints have different strengths that can complement each other, helping shore up weaknesses.
It was observed in chapter 3 that Lakoff’s focus on conceptual metaphor means that he neglects to spend much time considering the details of how metaphorical utterances relate
to concepts. Additionally, nowhere does Lakoff provide a account of fiction based in CMT.
Yablo’s Waltonian approach and its core metaphor provide a toehold for enriching this aspect
of the Lakovian theory, particularly his understanding of reference. Conceptual metaphors
and games of make-believe are posited to play the same role underpinning metaphorical utterances; the relationship between these two cognitive devices requires investigation.201 Yablo’s
clever handling of “the bomb” of self-applied number talk also provides CMT with resources
that could be used to fend off worries about abstract objects and circularity associated with
the concept of
CONCEPT;
while numerical mathematical sentences are his main focus, it is
worth noting that Yablo’s discussion aims to handle sentences purporting to refer to abstract
entities in general. On the other hand, Yablo’s focus on the linguistic side of things is certainly a contributing factor in his bemoaning the lack of clarity in the literal/metaphorical
distinction; understanding metaphor as conceptual can help clarify this situation. Finally,
wedding Yablo to Lakoff would be beneficial for the latter insofar as the relationship could
bring increased philosophical rigour and respect: Yablo is a respected and prolific member of
the contemporary philosophical community and, moreover, the idea that fiction is involved in
mathematics has been endorsed by many philosophers, including Bertrand Russell.202 Much
201
Elisabeth Camp’s arguments in “Two Varieties of Literary Imagination” distinguishing pretense from
seeing-as should be taken into account in any such future study.
202
“If there has been any truth in the doctrines that we have been considering, all numbers are what I call
logical fictions. Numbers are classes of classes, and classes are logical fictions, so that numbers are, as it
176
more work would be required to actually synthesize these two views in a fruitful way, and
there would likely be significant resistance from both sides; regardless, the potential gains to
be had from combining the Lakovian and Yablonian approaches makes further consideration
of this merger desirable.
At the end of chapter 3, it was suggested that incorporating semiotic insights could
help CMT overcome some of its most serious deficiencies. If semiotics has the potential to
bring balance to CMT generally, then it seems reasonable that it could make a beneficial
contribution to the special case of embodied mathematics as well. Brian Rotman has made a
career of “giving a semiotic analysis of mathematical signs,” and his work is not only coherent
with embodied mathematics but explicitly connects itself to many of the other mathematical
theories that have been considered in this chapter.203 His inquiry starts in a way nearly
identical to the one presented in this chapter: it explicitly understands mathematics as a
practice, and expresses dissatisfaction with the big three traditional theories of mathematics
while acknowledging that
to have persisted so long each must encapsulate, however partially, an important facet of what is felt to be intrinsic to mathematical activity. Certainly, in
some undeniable but obscure way, mathematics seems at the same time to be
a meaningless game, a subjective construction, and a source of objective truth.
The difficulty is to extract these part-truths: the three accounts seem locked in
an impasse which cannot be escaped from within the common terms that have
allowed them to impinge on each other.204
Observing that mathematical writing is permeated with imperatives — exhortations to “consider,” “define,” “add,” “integrate,” and so forth — Rotman asks what kind of actions result
from such requests given that “mathematics can be an activity whose practice is silent and
sedentary.”205 Drawing inspiration from Peirce, he concludes that the mathematical listener
were, fictions at two removes, fictions of fictions” (Russell, The Philosophy of Logical Atomism 111).
203
Brian Rotman, Mathematics as Sign: Writing, Imagining, Counting (Stanford: Stanford UP, 2000),
4. It is somewhat curious that relatively little work has been done in the semiotics of mathematics given
the prominent role of signs in that discipline. Moreover, the idea of applying semiotics to math is hardly
new. Over a century before Rotman, Peirce made multiple references to mathematics in his work on signs
(Rotman, Mathematics as Sign 4). Even earlier, Bolzano discussed the semiotics of mathematical symbols.
Interestingly, Bolzano foreshadows Lakatos, claiming that purposes precede definitions and “therefore it is
an error, contrary to good method, when Euclid gathers all his definitions at the beginning” (Bolzano 107).
204
Rotman, Mathematics as Sign 7.
205
Rotman, Mathematics as Sign 12. As mentioned in a previous note, Hoffman also emphasizes the role of
imperatives in mathematical practice in her dissertation.
177
responds to an imperative by engaging in an imaginative kind of thinking akin to a thought
experiment that is often inseparably amalgamated with “scribbling.” To understand these
interactions in a way coherent with the insights of the traditional theories, Rotman conceives
the agency of the mathematician as a trinity: Agent, Subject, Person. The Agent is an
automaton, a minimal, imaginary avatar of the mathematician that follows rules to perform
mathematical tasks; devoid of subjectivity, the Agent transcends finitude and other logical
constraints. The Subject is the master of the Agent, the authorial agency whose imaginings
form the worlds that the Agent is deployed into. As Rotman puts it, “our picture of the
Subject is of a conscious — intentional, imagining — subject who creates a fictional self, the
Agent, and fictional worlds within which this self acts. But such creation cannot, of course,
be effected as pure thinking: signifieds are inseparable from signifiers: in order to create
fictions, the Subject scribbles.”206 While the Subject is considerably more robust (though
less transcendental) than the Agent, her psychology is unsituated (“transcultural and disembodied,” in Rotman’s words) and entirely engaged with mathematics.207 The Person is the
embodied agency of the mathematician, capable of using natural language, constrained and
motivated by history, culture, and the practicalities of the world: the situated, unprojected
self.208 This conception of mathematical agency does a lot of work for Rotman, including
allowing for well-structured discussion of where each of the traditional theories goes wrong.
Note that, in terms of this trifurcation, both Lakoff and Lakatos believe that other theories
have problematically neglected consideration of the contributions of the Person to mathematics.209 Though this presentation of Rotman’s view is brief and incomplete, it is sufficient for
206
Mathematics as Sign 14.
Mathematics as Sign 15.
208
Rotman, Mathematics as Sign 15. Lakatos seems to endorse the Subject-Person distinction in the following passage:
207
mathematical activity produces mathematics. Mathematics, this product of human activity,
‘alienates itself’ from the human activity which has been producing it. It becomes a living,
growing organism that acquires a certain autonomy from the activity which has produced it; it
develops its own autonomous laws of growth, its own dialectic. The genuine creative mathematician is just a personification, an incarnation of these laws which can only realise themselves
in human action (146).
209
Rotman mirrors a major Lakatosian theme when he says “in the absence of the Person’s role, no explication of conviction — without which proofs are not proofs — can be given.” (Mathematics as Sign
54).
178
the purpose of briefly considering the coherences between his semiotic account and embodied
mathematics.
It is clear from the previous paragraph that, like Lakoff, Rotman focuses on mathematics
as practice. His approach is also similar to CMT in its inclusionary attitude: while Rotman
believes that the three traditional theories are incompatible and so a synoptic reconciliation
is impossible, he makes a concerted effort to understand each of them from the context of
his semiotic theory and incorporate their strengths where possible. He acknowledges that
metaphor seems to play an ineliminable role in mathematics, and also draws a connexion to
fictionalism in this significant passage:
any attempt to explicate mathematical thought is unlikely to escape the net of
. . . metaphors; indeed, to speak (as we did) of dwelling in a world of Hausdorff
spaces is metaphorically to equate mathematical thinking with physical exploration. Clearly, such worlds are imagined, and the actions that take place within
these worlds are imagined actions.210
Though they seem to disagree slightly about the details of the contribution, Rotman and
Lakoff and Núñez agree that gesture plays an important role in mathematical practice and
development.211 An intriguing but more tenuous connexion between CMT and Rotman comes
from Zoltán Kövecses, one of Lakoff’s collaborators. He distinguishes three levels on which
conceptual metaphors can be analyzed (the supraindividual, the individual, and the subindividual) which seem to roughly correspond with Rotman’s trinity of mathematical agency
insofar as they involve similar restrictions of scope.212 These coherences with Rotman offer
Lakoff a significant first step towards integrating semiotic insights into embodied mathematics, hopefully bringing balance to the theory by distributing focus across the semiotic triangle
rather than dwelling on the conceptual. Additionally, Rotman brings his own collection of
useful coherences with various mathematical theories that could help strengthen those of
Lakoff. It would be particularly interesting to compare and contrast the Subject-Agent account of infinite operations with the Basic Metaphor of Infinity, exploring the interplay and
synergies between these two approaches.213 While it would be great to incorporate semiotic
210
Rotman, Mathematics as Sign 12–3.
Brian Rotman, “Gesture in the Head: Mathematics and Mobility,” Mathematics and Narrative conference, Mykonos, 2005, 15–6. See also Núñez (2008).
212
Kövecses 239.
213
Lakoff and Núñez’s discussion of the FICTIVE MOTION metaphor provides a potential connexion to Rot211
179
insights into CMT at a more general level, the coherences between Lakoff and Rotman’s
theories of mathematics suggest that even this relatively restricted partnership could be very
fruitful.
What conclusions can be drawn from the above discussion? First, there is a good reason
that many contemporary theorists — including this author — fundamentally view mathematics as practice: whereas there is widespread disagreement about mathematical ontology, nearly everyone can agree that humans engage in an activity called mathematics and,
moreover, this activity is generally successful and beneficial. Further motivation for the
mathematics-as-practice perspective comes from its ability to bypass certain puzzles and
pitfalls that make the traditional theories untenable. While there are a variety of ways to
understand mathematical practices, a naturalistic approach that understands mathematical
activity as one among many human activities is appealing, as it roots itself in the vast wealth
of contemporary scientific results. The trick then becomes explaining the distinctive character
that seems to make mathematics unique among human practices: its universality, precision,
consistency, and so forth. One of the chief puzzles a naturalistic, practice-oriented approach
must address is what is going on when mathematicians seem to refer to abstract, empirically implausible entities (infinities, for example). A promising approach is to explain these
aspects of mathematics in terms of metaphor; in support of this idea, most people would concede that metaphors are clearly invoked in mathematical communication/instruction, such
as when Hilbert’s hotel is used to discuss the properties of infinity.214 Much of this document has aimed to provide support for the idea that metaphor plays an important role in
mathematics.
Discussion of Lakoff and Núñez’s embodied mathematics forms the core of this chapter because it is a naturalistic, practice-oriented theory that makes heavy use of conceptual
metaphor in explaining how even the most abstract and arcane results of mathematics are
man’s Agent that could help get this work underway (Lakoff and Núñez, “The Metaphorical Structure of
Mathematics,” 53–5).
214
David Hilbert used the idea of a hotel with infinitely many rooms in his lectures to explore the seemingly
paradoxical nature of infinity by showing that even if there is no vacancy and every room is full, the hotel
can still accommodate new guests — even an infinite number! (Hilbert, David Hilbert’s Lectures 730). Even
if people concede that metaphor is frequently invoked in mathematical discussion, this does not necessitate
that they accept it as a constitutive and ineliminable.
180
built up from basic human experiences and biological capacities. While the Lakovian perspective provides a promising starting point given the observations of the previous paragraph, the
theory is admittedly in its preliminary stages and is not without flaws; this chapter should not
be read as a wholehearted endorsement of Lakoff. There is definitely room for improvement
and development in the Lakovian approach; accordingly, several other promising contemporary approaches to mathematics were also considered. However, the Lakovian approach
has one additional compelling property: CMT is not only a theory of mathematics, but is
a general theory of concepts and, therefore, is self-applicable; that is, Lakoff gives us not
only a theory of mathematics but also a theory of theories of mathematics. While the traditional theories of mathematics were all rejected as flawed, each was also seen to capture some
important aspect of mathematical practice. Metaphor is a device that allows one to bring
useful correspondences to the fore while obfuscating problematic and disanalogous aspects
and, using conceptual metaphor, Lakoff and Núñez incorporate the fundamental insights of
the traditional theories into embodied mathematics. Thus, while reconciliation with rival theories was seen as impossible from within any of the traditional perspectives, Lakoff provides
a mildly relativist framework within which even radically contradictory understandings can
legitimately coexist to some extent. Lakoff and Núñez explicitly incorporate important aspects of platonism and formalism into embodied mathematics as conceptual metaphors; this
dissertation continues this trend by establishing coherences between embodied mathematics
and some rival contemporary theories, and by speculating about the core metaphors that may
constitutively underlie these rivals. Lakatosian quasi-empiricism, Yablonian figuralism, and
Rotmanian semiotics all have potential to enrich embodied mathematics and help it overcome
some of its key deficiencies, including accounting for the historical and symbolic/linguistic
aspects of mathematics, whether they are found to be outright consistent with the Lakovian
approach or only metaphorically coherent. Much work remains if the suggested patchwork
theory is to become a reality.
It must be remembered that the theorizing in this chapter is primarily descriptive rather
than prescriptive. It recognizes the amazing success of most mathematical practices, and
does not seek to alter them by, say, forbidding reference to abstract entities or reducing the
level of formal rigor. The prescriptions it does make are primarily aimed at philosophers
181
of mathematics, suggesting that the paradigm shift to a practice-oriented approach is legitimate and useful, for example. There is, however, one aspect of mathematical practice
that should be changed: the distaste and crippling confusion experienced by a significant
proportion of people when they engage with mathematics. While the highly formalized
mathematics performed by professional mathematicians is very effective in communicating
to the mathematically adept, it will generally be ineffective in communicating to the layperson. A prototypical version of such communication occurs between mathematics teachers
and their pupils. The recognition that conceptual metaphors underlie mathematical reasoning provides potential inspiration for alternative modes of communication and instruction
that are more effective and avoid contributing to the widespread dislike of mathematics. It
is perhaps unsurprising that schoolteachers are perhaps the most vociferous supporters of
Where Mathematics Comes From, and that novel approaches to teaching mathematics based
in conceptual metaphor are now occasionally implemented.215 It would be nice to see trials of
a conceptual-metaphor-based approach in the post-secondary mathematics classroom, where
both professors and students are often dissatisfied with the educational process.
The main goal of this dissertation is to investigate the relationship between mathematics
and metaphor. Thus far, the focus has been on the metaphor side of the equation, considering
what role, if any, metaphor plays in mathematics. Answering this question even somewhat
satisfactorily required a lengthy discussion about the nature of metaphor, ultimately concluding that contemporary understandings of metaphor as fundamentally conceptual rather
than merely linguistic are superior. Proceeding from that conclusion, the present chapter has
argued that conceptual metaphor helps explain not only mathematical practice but also the
philosophy of mathematics, providing a mechanism for amalgamating the positive aspects of
various theories of mathematics while avoiding contradiction. The final core chapter of this
dissertation goes against the flow of the previous three chapters and investigates the extent
to which metaphor can be understood in terms of mathematics.
215
See, for example, Danesi (2007).
182
Chapter 5
Metaphor is Mathematical
Every discourse on metaphor originates in a radical choice. . . If it is
metaphor that founds language, it is impossible to speak of metaphor
unless metaphorically. Every definition of metaphor, then, cannot but
be circular. The other option is that metaphor is a deviation from
normal, preestablished rule-governed language.1
— Umberto Eco
The modeling capacity of mathematics is astounding. Humans constantly understand the
world in mathematical terms, from counting the bottles of beer left in the fridge to describing
the properties of subatomic particles. While some experiences are perhaps not well-suited to
mathematical explanation, there is very little that is entirely beyond the purview of mathematics.2 Almost all empirical science, for example, involves some quantitative aspect. It is
clear at any rate that language and communication are prime candidates for mathematical
modeling. An important question thus arises: to what extent can metaphor be understood
in terms of mathematics?
There are three possible answers to this question: not at all, somewhat, and entirely.
The first answer is not only uninteresting and would make for a very short chapter, but also
seems demonstrably false given the number of researchers making progress in this area. The
third answer is what many of the aforementioned researchers are aiming for, but attaining a
1
2
Eco, Semiotics and the Philosophy of Language 88.
As John Barrow puts it, foreshadowing some of my upcoming sentiments about applied math:
Mathematics is also seen by many as an analogy. But it is implicitly assumed to be the
analogy that never breaks down. Our experience of the world has failed to reveal any physical
phenomenon that cannot be described mathematically. That is not to say that there are not
things for which such a description is wholly inappropriate or pointless. Rather, there has yet to
be found any system in Nature so unusual that it cannot be fitted into one of the strait-jackets
that mathematics provides (Barrow 21).
183
robust mathematical model for generating and interpreting metaphors seems a long shot given
the complexity and elusiveness of the subject matter. Eco agrees with and surpasses this
sentiment: “No algorithm exists for the metaphor, nor can a metaphor be produced by means
of a computer’s precise instructions, no matter what the volume of organized information to
be fed in.”3 Whatever one’s opinion on whether this ambitious goal can possibly be achieved,
it is certainly beyond the expertise of this author and the scope of this document.
The primary goal of this chapter is to provide a partial understanding of metaphor
using mathematical metaphors. While authors from Nicomachus to Lakoff and Johnson
have claimed that mathematical modeling and application are metaphorical in nature, the
metaphors invoked in this chapter will be of a less rigorous variety.4 That is, while this chapter will invoke a variety of mathematical concepts — some which may be unfamiliar to those
with no advanced mathematics training — it should not be viewed as a work of mathematics.
The imprecise use of precise concepts and theories is distasteful to many specialists, but does
not seem inherently problematic if the imprecision is acknowledged. Though no mathematical models for metaphor will be offered in this chapter, it is possible that the metaphors
introduced here could one day be developed into something more rigorous and exact. Any
such developments would be welcome, but the understanding of metaphor presented in this
chapter is meant to be useful in its own right, not merely as a launching pad for precise
mathematization.
5.1
Computational Linguistics
For the sake of completeness, it is worth briefly considering the history of attempts to understand language and metaphor mathematically; readers who find the metaphorical approach
distasteful may find something more palatable in this section. The connexions between mathematics and language extend far back into human history: because language provides a rich
source of patterns, and because mathematics is aptly described as “the science of patterns,”
it seems inevitable that mathematics would be applied to language structure.5 Language pat3
Umberto Eco, “Metaphor, Dictionary, and Encyclopedia,” New Literary History 15.2 (1984): 269.
Athenaeus 290; Lakoff and Johnson, Philosophy in the Flesh 91.
5
Keith Devlin, Mathematics: The Science of Patterns (New York: Scientific American Library, 1997), 65.
4
184
terns are perhaps most blatantly apparent in poems, linguistic works observing structuring
constraints on metre, rhyme scheme, verse shape, and so forth; the creation of many forms
of poetry involves numerical considerations, particularly in the form of syllable counting in
styles as diverse as haiku and iambic pentameter. Another connexion between language and
mathematics is that, prior to the introduction of Hindu-Arabic numerals to Europe around
1000 CE, glyphs from the Hebrew, Greek, and Latin alphabets did double duty, serving both
as letters and as numerals.6 Most readers will be somewhat familiar with this phenomenon
from studying Roman numerals in grade school (I=1, V=5, X=10, L=50, and so on), even
if this connexion was not explicitly recognized previously. This association between numbers and letters was exploited in numerological practices which sought to understand words
by way of the numerical result of summing the values associated with their constituent letters; this was called gematria in the Judiac tradition and isopsephy in the Greek.7 Today,
numbers are still associated with letter symbols, though for less spuriously arcane purposes.
ASCII, Unicode, and other encoding schemes associate alphabetic symbols with numerical
codes that allow computers to handle text.8 While every personal computer is capable of
decoding ASCII, other numerical ciphers are employed when sensitive messages must be conveyed securely. While these symbol-oriented connexions between numbers and language are
interesting, and set the stage for higher-level modeling, they do not help with understanding
metaphorical phenomena.
There have been many attempts to mathematize higher-level syntactic and semantic aspects of language. Early attempts typically aimed to reduce confusion and conflict, appealing
to mathematics to introduce precision and clarity to language; these efforts were generally
prescriptive rather than descriptive and, if successful, probably would have involved the elimination of metaphor from most discourse. Aristotle’s development of syllogistic logic in the
Organon is arguably the first serious attempt to present a systematic theory of reasoning in
6
Barrow 93.
Ivor Grattan-Guiness, The Norton History of the Mathematical Sciences (New York: W.W. Norton and
Company, 1997), 125. The “Revelation” that the Number of the Beast is 666 is perhaps the most widely
familiar example of gematria.
8
TEX, the typesetting system used to produce this document, works somewhat differently from these
encoding schemes, but still ultimately must use numbers to encode features of text.
7
185
order to avoid error and sophistry.9 In the Discourse on Method, Descartes holds the evidentness of mathematical proofs as the ideal that all other reasoning should aspire to: “Those
long chains of utterly simple and easy reasonings that geometers commonly use to arrive
at their most difficult demonstrations had given me occasion to imagine that all the things
that can fall within human knowledge follow from one another in the same way.”10 This
line of thinking culminates in Leibniz’s project to develop the characteristica universalis, a
symbolic universal language combining the scope of natural language with the precision and
calculability of arithmetic and algebra:
If we had some exact language . . . or at least a kind of truly philosophical writing,
in which the ideas were reduced to a kind of alphabet of human thought, then all
that follows rationally from what is given could be found by a kind of calculus,
just as arithemetical or geometrical problems are solved . . . I can demonstrate
with geometric rigor that such a language is possible, indeed that its foundation
can be easily laid within a few years by a number of cooperating scholars.11
In such a language, disagreements would be resolved not by mediation or compromise, but
by calculation: “when a controversy arises, disputation will no more be needed between two
philosophers than between two computers. It will suffice that, pen in hand, they sit down to
their abacus and . . . say to each other: let us calculate.”12 Central to this project is a conviction that all human ideas are made up from a set of primitive concepts, just as every integer
has a unique prime factorization.13 By directly symbolizing these primitives as naturally as
possible (drawing on inspiration from Chinese characters and Egyptian hieroglyphics) and
providing precise mechanisms of composition and derivation, Leibniz hoped to create an external isomorphic copy of our conceptual system that would eliminate all errors of reasoning
apart from the occasional simple calculation mistake.14 In places, Leibniz proposes using
9
Robin Smith, “Aristotle’s Logic,” Stanford Encyclopedia of Philosophy, 28 Jan. 2014.
René Descartes, Discourse on Method, Trans. Donald Cress, 3rd ed. (Indianapolis: Hackett, 1998), 54.
11
Gottfried Wilhelm von Leibniz, “On the Universal Science: Characteristic,” Monadology and Other
Philosophical Essays, Ed. and trans. Paul and Anne Martin Schrecker (New York: Bobbs-Merrill, 1965), 12;
emphasis his.
12
Leibniz 14; emphasis his. Of course, the use of the word “computers” in this translation refers to the
profession and not the electronic devices ubiquitous in present day, though making such an anachronistic
mistake seems to make little difference in this particular case. Leibniz’s stepped reckoner was the first
mechanical calculator (theoretically) able to perform all four arithmetic operations, and thus was an important
predecessor of the electronic computer (Barrow 128).
13
There could be interesting connexions between Leibniz’s primitives and Lakoff’s basic-level concepts that
are worth exploring.
14
Leibniz 18.
10
186
prime numbers rather than representationally faithful logograms to represent the primitive
concepts, and representing compound concepts with characteristic numbers formed by taking
the product of their primitives — an approach very similar to that implemented by Gödel in
proving his incompleteness theorems.15 Elsewhere, Leibniz suggests that the characteristica
universalis would have to be able to accommodate probabilistic considerations, anticipating
to some small extent contemporary statistical linguistic models.16 Though Leibniz apparently
abandoned this project, his suggestions were ahead of their time and importantly influenced
later attempts to mathematize language.
The nineteenth century saw several important developments that led to a change in the
goals of language mathematization projects, including significant work in axiomatic systems,
formal symbolic logic, mechanical computation and communication devices, and linguistics.
Emphasis began to split between trying to develop new mathematically precise languages
oriented from concept-to-sign and attempting to mathematically describe extant natural
language use from sign-to-concept; rather than trying to eliminate error by artificially constraining language, researchers attempted to include the double-edged complexities of natural
language that generate both errors and richness into their models. These efforts to mathematically describe the mechanisms of language importantly have application in machinetranslation and artificial-intelligence research; the explosive technological development of the
twentieth century correspondingly induced a significant increase in linguistic modeling. Models of language have become increasingly technical and sophisticated, and yet many of them
belong to the tradition that marginalizes metaphor as derivative and deviant and attempts
to eliminate it through paraphrase. Some of the most promising contemporary approaches
— such as the one underlying Google Translate — break with tradition and use statistical
rather than algorithmic models, relying on unfathomably massive collections of data and
blistering computing speed to mechanically generate sentences.17 Such models may be able
15
Barrow 128. Interestingly, even though his proof that some propositions are undecidable seems to constitute a major problem for Leibniz’s project, Gödel maintained that the characteristica universalis was viable
and, furthermore, that a conspiracy systematically suppressed publication of the portion of Leibniz’s oeuvre
dedicated to developing the universal language (Dawson 166). Given that much of Leibniz’s writing remains
unpublished, it is conceivable that important results yet lurk undiscovered in the archives.
16
Leibniz 15.
17
“Inside Google Translate,” Google Translate, 28 Jan. 2014.
187
to translate and generate linguistic metaphors, yet do not seem to contribute much to understanding metaphor.18 However, this should not be understood to imply that no attempts
to model metaphor have been made.
Several researchers have specifically attempted to mathematically model metaphorical
reasoning; such authors tend to subscribe to some non-traditional view of metaphor-asconceptual. As discussed above, the details of these theories are beyond the scope of this
chapter and the expertise of this author: I am no computational linguist or informaticist.
A brief list of a few noteworthy efforts follows for those readers interested in pursuing a
more-precise mathematical approach to metaphor than is presented below. Perhaps the best
known and most widely implemented model is Dedre Gentner’s Structure Mapping Engine
(SME), an algorithmic model of analogy that emphasizes higher-order structural similarities
over shallow similarities; Gentner conceives of metaphor as invoking analogical reasoning.19
A variety of competing predictive models are offered by Keith Holyoak and his coauthors:
ACME (Analogical Constraint Mapping Engine), ARCS (Analogical Retrieval by Constraint
Satisfaction), CWSG (Copy With Substitution and Generation), LISA (Learning and Inference with Schemas and Analogies), and ECHO (Explanatory Coherence by Harmony
Optimization).20 Interestingly, the more recent effort BART (Bayesian Analogy with Relational Transformation) integrates a probabilistic component.21 Douglas Hofstadter and his
Fluid Analogies Research Group criticize Gentner, Holyoak, and their various contemporaries
for bypassing representation building in their models: “if appropriate representations come
presupplied, the hard part of the analogy-making task has already been accomplished.”22
The models generated by the Fluid Analogies Research Group — Seek-Whence, Jumbo,
Numbo, Copycat, Tabletop, and Letter Spirit — accordingly focus on avoiding this criti18
This is not entirely fair. Even with my extremely limited understanding, it seems clear that such models
must include a mechanism for sifting through the data for similar or analogous cases; studying the mechanisms
employed for this purpose could be extremely enlightening.
19
See Gentner (1983), Falkenhainer, Forbus, and Gentner (1989), and Gentner et al. (2001) to start; there
are many, many publications on SME if one is compelled to investigate this theory at length.
20
Keith J. Holyoak, “Overview of Research Career,” Keith Holyoak’s Web Page, UCLA Reasoning Lab,
18 Aug. 2013. For more information, see his extensive publication list on his website; the original paper is
Holyoak and Thagard (1989).
21
See Lu, Chen, and Holyoak (2012).
22
David Chalmers, Robert French, and Douglas Hofstadter, “High-level Perception, Representation, and
Analogy,” Fluid Concepts and Creative Analogies (New York: Basic Books, 1995), 182.
188
cism.23 Though there are many additional metaphor modeling projects currently under way,
one other stands out as particularly relevant here: a team of informaticists from the University of Edinburgh is working on a computational model based in Information Flow theory
with the specific aim of modeling Lakoff and Núñez’s arithmetic grounding metaphors.24
While such modeling projects are interesting and admirable, their details are outside the
purview of this chapter.
There are several reasons that this chapter provides only a cursory list of these modeling
projects. First, as mentioned above, much of the work being done is outside of my current
scope of expertise: I am not in a position to summarize, let alone evaluate, their efforts. Second, several noteworthy scholars have concerns that all such modeling projects are inherently
flawed. Lakoff’s complaint seems to stem from the fact that functions, by definition, can have
at most one output per input: “the contemporary theory of metaphor is at odds with certain
traditions in symbolic artificial intelligence and information processing psychology. . . those
traditions must characterize metaphorical mappings as an algorithmic process, which typically takes literal meanings as inputs and gives a metaphorical reading as an output. This
runs counter to cases where there are multiple, overlapping metaphors in a single sentence,
and which require the simultaneous activation of a number of metaphorical mappings.”25 The
details of Eco’s objection are a bit obtuse, being couched in technical semiotic jargon, but his
conclusion is crystal clear (and thus worth repeating): “No algorithm exists for the metaphor,
nor can a metaphor be produced by means of a computer’s precise instructions, no matter
23
See Hofstadter and the Fluid Analogies Research Group (1995). One regret I have in writing this
dissertation is that, for various reasons, I have not been able to incorporate more of Hofstadter’s work. Much
of his thought on metaphor and analogy seems coherent (and even consistent!) with the Lakovian perspective
and — more importantly — with my own. He unabashedly claims that “every concept we have is essentially
nothing but a tightly packaged bundle of analogies” (Hofstadter, “Epilogue” 500), and, correspondingly, that
“analogy-making lies at the heart of intelligence” (Hofstadter, Fluid Concepts and Creative Analogies 63). He,
like me, started as a graduate student in mathematics but ended up changing disciplines when he discovered
that “I had always thought that I was a pretty abstract thinker, but. . . in fact, all of my thoughts are very
concrete. They are all based on images, analogies, and metaphors. I really think only in concrete ideas, and
I found that I couldn’t attach any concrete ideas to some of the mathematics I was learning.” (Hofstadter,
“Analogies and Metaphors to Explain Gödel’s Theorem” 98). Had things gone only slightly differently during
the conception of my project, Hofstadter could well have formed the core of this dissertation instead of Lakoff;
I will certainly be integrating more of his work into my future projects.
24
See Guhe, Smaill, and Pease (2009).
25
Lakoff, “The Contemporary Theory of Metaphor” 249.
189
what the volume of organized information to be fed in.”26 Whether these concerns are wellfounded is of minor relevance here. Even if a thorough and robust computational model of
metaphorical reasoning and language were obtained, it would not thereby constitute the only
legitimate approach to understanding metaphor! While a precise algorithmic understanding
would be ideal for programming computers to generate and/or interpret metaphors, it seems
unlikely that a thorough computational model would be concise or transparent, making it
less beneficial for aiding the understanding of most human individuals. A good metaphor,
on the other hand, can efficiently and effectively capture the essence of a complex idea and
simplify communication. For example, while “Crotone is located at 39◦ 05′ N 17◦ 07′ E” is
precise, it is also theoretically-dense and therefore inaccessible to someone unversed in the
language of the global coordinate system, whereas Walton’s “Crotone is on the arch of the
Italian boot” conveys immediate understanding to anyone who has the most basic knowledge
about the globe and footwear.27 Although both kinds of understanding are valuable, it is the
imprecise but succinct metaphorical variety that this chapter aims to generate.
5.2
Mathematical Metametaphors
In chapter 2, it was shown that from its very origins,
METAPHOR
has itself been understood
by way of metaphors — what I have been calling metametaphors. The remainder of this
chapter is dedicated to developing metametaphors with mathematical source domains. A
brief defense and justification of this approach is warranted. At the beginning of the chapter,
it was noted that some mathematically inclined specialists may find the imprecise application
of mathematical concepts distasteful and problematic; however, the explicit acknowledgment
that this non-standard use of mathematical concepts is not intended to be understood as a
work of mathematics seems to vindicate its relative lack of rigor and precision. In particular,
this approach should not be taken as endorsing — or even condoning — sloppy, inconsistent
mathematics. Consider the following analogous case: intentionally and deliberately using
metre sticks to measure weight is an atypical practice that employs a tool for a purpose other
26
27
Eco, “Metaphor, Dictionary, and Encyclopedia” 269.
Walton, “Metaphor and Prop Oriented Make-Believe” 40.
190
than the one it was devised for.28 However, such an act of weighing is not invalidated simply
by virtue of its atypicality, and certainly does not constitute an unskilful or “sloppy” act of
measuring length! A related concern is that employing metaphor to understand metaphor
leads to vicious circularity. However, the kind of circularity involved does not seem to be
necessarily problematic. For example, a micrometer-manufacturing machine may have its
dimensions and tolerances measured by one of the micrometers it has itself created. Insofar
as all human understanding involves the brain, our understanding of neurons is neuronal.29
Gödel numbers encode arithmetic expressions as integers, thereby, in a sense, using arithmetic
to understand arithmetic. So it is for metaphor:
Every scientific theory is constructed by scientists — human beings who necessarily use the tools of the human mind. One of those tools is conceptual metaphor.
When the scientific subject matter is metaphor itself, it should be no surprise
that such an enterprise has to make use of metaphor, as it is embodied in the
mind, to construct a scientific understanding of what metaphor is.30
Thus, there seems to be a sufficient gap between metaphor qua capacity and metaphor
qua empirical phenomenon to avoid viciousness, though extra vigilance in watching out for
question-begging is warranted. Finally, I concede that the metametaphors developed below
may be inaccessible to a large proportion of the population, because the mathematical ideas
invoked may lie beyond the competency of the average person, and because those who do have
an understanding of the mathematical ideas involved are more likely to have an aversion to
their being used in this way.31 Worries that a given approach may not be widely accessible or
popular should not be considered an adequate reason to abandon inquiry; while I hope that
at least some readers find the ideas presented in this chapter useful or at least interesting,
that I find them useful and interesting provides sufficient grounds to proceed.
The use of mathematical metametaphors is not unprecedented. Lakoff and Johnson say
that their original conception of conceptual metaphor relied on the metaphor
METAPHORS ARE
MATHEMATICAL MAPPINGS:
28
For example, at the time of writing this, I weigh 653 metersticks.
Though they are making a different point, this example was inspired by Lakoff and Johnson (Philosophy
in the Flesh 103).
30
Lakoff and Johnson, Metaphors We Live By 252.
31
Of course, people committed to a traditional conception of metaphor as an eliminable linguistic garnish
are also unlikely to find this work particularly compelling.
29
191
Our first metaphor for conceptual metaphor came from mathematics. We first
saw conceptual metaphors as mappings in the mathematical sense, that is, as
mappings across conceptual domains. This metaphor proved useful in several
respects. It was precise. It specified exact, systematic correspondences. It allowed for the use of source domain inference patterns to reason about the target
domain. Finally, it allowed for partial mappings. In short, it was a good first
approximation.32
However, they eventually saw METAPHORS ARE MATHEMATICAL MAPPINGS as problematically disanalogous, failing to capture a crucial feature of metaphor: namely, that “[m]athematical
mappings do not create target entities, while conceptual metaphors often do.”33 Lakoff and
Johnson went on to explore other metametaphors, including as METAPHOR IS SLIDE PROJECTION,
before adopting their current neural approach to metaphor.34 However, their abandonment
of
METAPHORS ARE MATHEMATICAL MAPPINGS
seems premature. Mathematics does have some
resources for dealing with the creation of new entities which may circumvent Lakoff and
Johnson’s concerns; moreover, even if mathematics did not possess any such resources, this
would seem to indicate a promising direction for future mathematical development rather
than a dead end. The metaphor
METAPHORS ARE MATHEMATICAL MAPPINGS
has more to offer
than Lakoff and Johnson realized; a more detailed exploration of some mathematical examples and their metaphorical entailments will show that this metametaphor still has the
potential to make an important contribution to how conceptual metaphor is conceived.35
The plan for the remainder of this chapter is to consider some mathematical analogues of
the core salient properties of metaphor that were discussed earlier in the dissertation, thereby
hopefully rejuvenating and improving upon the Lakovian mathematical metametaphor in the
process. One point that everyone from Aristotle to Lakoff agree on is that metaphor involves
a comparative or interactive relationship between two things or ideas; this most basic observation is at the heart of the
METAPHORS ARE MATHEMATICAL MAPPINGS
metaphor. However,
mathematical mappings come in a variety of flavours, some of which are a better fit for
32
Metaphors We Live By 252.
Lakoff and Johnson, Metaphors We Live By 252.
34
Lakoff and Johnson are not the only ones to invoke mathematical metametaphors: recall Yablo’s logarithm analogy for understanding games of make-believe, including metaphor (“Go Figure” 182).
35
It should be noted that Lakoff tends to focus his attention on entrenched “metaphors we live by” that are
constitutively integrated into our conceptualizations. The metaphors offered below are creative and novel,
not ubiquitous and conventionalized. That is, the following metaphors have been invented to help make sense
of the difficult notion of METAPHOR, not discovered as an existing part of the common understanding.
33
192
metaphor than others. First, because we understand metaphors as inference-preserving, it
seems metaphors are more analogous to morphisms — structure-preserving mathematical
mappings — than to simple binary relations; the prototypical morphism is the function,
but the class also includes homomorphisms and functors.36 While mere relations are too
unstructured to be a good fit for metaphor, other mappings are too structured. Metaphors
necessarily cross domains and involve disanalogy, and are therefore not identity mappings;
more generally, the disanalogy requirement suggests that only a proper subset of the source
domain is mapped by the metaphor. There is neither a requirement that a metaphor subsume every part of the target domain, nor a prohibition against distinct components of the
source domain mapping to the same component of the target domain; metaphors are therefore
typically neither surjective nor injective. The asymmetry noted in many metaphors manifests mathematically as non-invertibility of the mapping. Even the functionhood of some
metaphor maps might be questioned, as the same metaphor could be interpreted differently
by different people (or by the same person at different times), meaning a single element of
the source domain might map to multiple targets; to keep this discussion fruitful and accessible, metaphorical mappings will be assumed to be functions.37 Thus far, more has been
said about what metaphors are not than what they are; in general, their inference-preserving
nature means they are best understood as functions or, perhaps, homomorphisms. To delve
further, we must consider what metaphors map between.
Conceptual metaphors connect two concepts, or conceptual domains. Mathematical maps
connect two mathematical objects or structures. The question is, what mathematical structures are usefully analogous to concepts? Discussion in chapter 3 showed that humans employ
a broad and diverse range of concepts, and therefore providing a theory that adequately accounts for them all is a staggeringly difficult task. It thus seems unlikely that any metaphor
36
The main difference between these morphisms is what kind of mathematical objects they map between.
For example, group homomorphisms are operation-preserving mappings between groups, real functions are
structure-preserving mappings on the set of real numbers, and functors are morphism-composition-preserving
mappings between the abstract structures known as categories.
37
One might try to preserve functionhood by considering different interpretations as different mappings, or
by exploiting the lack of simultaneity, or by attaching probabilities to the various targets. These are promising
approaches, but it nonetheless seems possible that a single interpretation could simultaneously map a single
source to multiple targets, though no example springs to mind. There is a theory of multifunctions that
could be applied here if warranted.
193
provided here will be able to do more than capture some generalities about concepts. First,
however, consider that CONCEPTS ARE INTEGERS does not seem to be an especially apt metaphor
despite a certain amount of historical popularity. As discussed above, Leibniz’s characteristica universalis project aimed to find “conceptual primes” that could be combined into
more-complex concepts via multiplication. Conceptual atomists adopt a similar stance — recall from chapter 3 that they hold that, just as unfactorable prime numbers can be multiplied
to make composite numbers, concepts are structureless primitives that can be composed into
complexes — but without Leibniz’s emphasis on computation. However, the posit of primitive concepts with no internal structure is incompatible with the understanding of concepts
found in CMT, where even the most basic, image-schematic concepts have innards.38 The
idea that concepts themselves are structured is certainly not unique to CMT; for example,
Douglas Hofstadter makes the claim that “words and concepts are far from being regularly
shaped convex regions in mental space; polysemy (the possession of multiple meanings) and
metaphor make the regions complex and idiosyncratic.”39 It thus seems that something more
mathematically complicated than numbers is needed here: at the very least, insofar as we
understand conceptual metaphors as structure-preserving mappings, the objects they map
between should possess some structure to preserve.
Though they are the first (and possibly only) kind of mathematical objects to come to
mind for most people, numbers are far from the only mathematical entities available for
consideration. In Metaphors We Live By, Lakoff and Johnson claim that many concepts
possess a unified multidimensional structure that we impose on our experiences in order to
classify and understand them.40 To the extent that this is deemed plausible, it makes sense
to understand a concept as a structure existing inside some multidimensional space. Why not
understand the concept as the entire multidimensional space? There are at least two reasons.
One, concepts are generally not static totalities but tend to change, grow, and develop over
time, potentially even expanding into new dimensions; in particular, if metaphors are to be
able to create target entities as Lakoff insists, they must have some uncharted room to do
so in. Positing that concepts are contained in infinite spaces ensures that this potential for
38
Lakoff, Women, Fire, and Dangerous Things 279.
“Epilogue” 511.
40
Metaphors We Live By 80–3.
39
194
development is never exhausted. Two, in some sense a proper subset of an infinite space can
actually possess more interesting structure than the space itself insofar as the former has
additional constraints. This observation also coheres with Hofstadter’s claim about concepts
possessing complicated shapes. While an entire infinite space such as Rn is arguably too
big, complete, and symmetrical an entity to represent a concept, singleton points or vectors
within such a space are also likely inapt insofar as they are are nearly as limited in structure
as numbers. One more general observation about conceptual space is salient here. Even
though it seems clear that concepts can possess vastly different amounts of complexity and
dimensions of structure, and even though concepts like
COFFEE, TELEPORTATION,
and
FEELINGS
seem to be distinctly distinct, there are good reasons to think of all of a person’s concepts
as existing within a single conceptual space. First, this understanding avoids multiplying
entities unnecessarily: it is possible to imagine a single space that is vast and dimensioned
enough to accommodate them all. Second, it seems rather natural to consider a person’s
concepts to exist within a single space insofar as they exist within a single mind. And third,
understanding concepts as occupying a single space is coherent with our conceptualization of
concepts as being more or less distant from each other:
to
DUCK
than it is to
AARDVARK
or
GOOSE
is usually thought of as closer
ADVERB.
This observation suggests that conceptual space may be a metric space; that is, that
a “distance” function is defined on it. Such structure could be used to help explain the
typicality effects observed by Rosch:
BIRD
than
ROBIN
PENGUIN
could be understood as more distant from
is, for example. However, there are also reasons to think that the notion
of conceptual distance does not possess all of the necessary definitional characteristics of a
metric. As a side effect of the typicality gradient, one might see
PENGUIN
as closer to
ROBIN
than ROBIN is to PENGUIN, for example, because, as it were, the first distance assessment takes
the measurer deeper into the concept BIRD while the second measurement leads one out to its
periphery. Thus, unlike a metric, conceptual distance seems to lack symmetry with respect to
its arguments. For similar reasons, it is not clear that the triangle inequality holds; depending
on how addition of “distances” is interpreted,
ROBIN
and
PENGUIN
might plausibly be seen as
closer together if one goes by way of BIRD instead of directly.41 While it seems uncontroversial
41
The triangle inequality: if d is a metric on a space X then d(x, y) ≤ d(x, z) + d(z, y) ∀ x, y, z ∈ X.
195
that there is no distance between a concept and itself, there may also exist cases where two
distinct concepts are judged to have zero distance between them, at least within a context
where the differences are irrelevant to the purpose at hand: the distance between
GOOSE
and
WILD GOOSE
DOMESTIC
might be zero at lunchtime, though this is a controversial claim. Of
course none of this is definitive; despite these observations, it may be possible through careful
mathematical work to construct a well-defined conceptual metric. For example, there may
be multiple notions of distance at work here that could be teased apart. Even if the idea
of conceptual distance cannot be made well-behaved enough to be considered a full-fledged
metric, this does not mean it is not useful or even that it is mathematically illegitimate
— it might simply be a restricted kind of distance that fails to satisfy one or more of the
conditions of metrichood, or it could induce a topology on conceptual space, or it could do
none of these things and still be an interesting property of the space. In any case, conceptual
distance is one kind of structure that could be preserved by metaphorical mapping. Enough
about conceptual space, on to concepts proper.
So far, concepts have been described as internally structured, complicatedly shaped entities coexisting within a multidimensional infinite space endowed with some kind of “distance”
function. Additionally, it was noted that concepts are not static entities but rather are dynamic, changing and developing over the course of a person’s life. Each successive version
of a particular concept depends on a variety of factors, including experiential inputs, other
conceptual developments, and the previous version of the concept.42 This observation suggests a very promising metaphor:
CONCEPTS ARE FUNCTIONS.
If concepts are functions, then
conceptual space is a function space, a vector space whose elements are functions.43 Concepts
can thus be seen as both individual points of a space as well as entities possessing shape and
internal structure in their own right, certainly far more structure than an individual n-tuple;
42
It is worth noting that the word “version” has connotations of discreteness, but it is probably best to
understand concepts as being continuously dynamic.
43
Actually, given the idea that concepts develop and change over a person’s lifetime, we should probably
think of concepts as time-dependent paths through a function space. Moreover, these trajectories should
probably be treated as dynamical systems insofar as each successive version of the concept significantly
depends upon the previous one. However, for the rest of this chapter, we will primarily think of concepts as
single elements in a function space for two reasons. First, as mentioned in chapter 4, CMT has a synchronic
bias. Second, this will simplify the mathematics considerably, hopefully allowing this chapter to remain at
least somewhat accessible.
196
indeed, functions are interpreted set theoretically as sets of ordered pairs (that is, 2-tuples)
of the form (input, output). Drawing inspiration from Hofstadter’s idea of conceptual shape,
recall that most of the functions a person deals with in their early mathematical training
can be represented by plotting the set of ordered pairs to form a graph of the function; for
example, the function f : R → R defined by f (x) = x2 can be represented in the Cartesian
plane as a parabolic curve opening upward whose vertex is located at the origin. In principal, the idea of graphing can be extended to functions defined on more complicated domains
and codomains, though creating adequate visual representations of such graphs may be impossible due to the constraints of the Euclidean 3-space we inhabit.44 Having understood
concepts as functions — and, in particular, as the graphs of those functions — it makes sense
to next ask what the domain and codomain of concepts are. Here, I take a metaphorical
liberty and assume them to both be R, even though this conflicts with the understanding
of concepts as multidimensional (Lakoff and Johnson’s analysis of
WAR
requires at least six
dimensions, for example).45 There are good reasons for doing this. Concepts are only being
addressed here because they form a necessary component of the understanding of metaphor
under consideration. Given that
CONCEPT
is a massively complicated and difficult concept to
understand, it would be easy to get caught up in trying to create an increasingly detailed
metaphorical understanding of this peripheral notion, thereby losing sight of the primary
objective of this discussion. Imposing the real numbers as the domain and codomain of concepts (and intentionally leaving these axes uninterpreted) allows focus to return to metaphor.
While choosing a domain with more dimensions would be more faithful to Lakoff (and to
my own intuitions), the simpler choice made here is communicatively preferable as it makes
the necessary mathematics more accessible and allows the reader to visualize functions in a
familiar setting. In principle, interested parties should be able to generalize the discussion
below to a variety of different domain-codomain pairs that would be more apt.
44
Let f : X → Y be a function. The domain of f is X, the set of inputs for which f is defined. The
codomain of f is Y . The image or range of f is f (X), the set of all outputs of f . The image of a function is
not necessarily identical to its codomain, though f (X) ⊂ Y .
45
Metaphors We Live By 80–1.
197
If concepts are functions, then metaphors map functions to functions in a structurepreserving way.46 To help gain some insight into what the nature of this mapping could
be, recall that metaphor is frequently conceived as an act of seeing-as: seeing Italy as a
boot, seeing Achilles as a lion, “seeing” (experiencing) affection as warmth, and so on. Two
situations in mathematics that could be understood as seeing-as spring to mind. First, in
linear algebra, points in R3 are usually described as linear combinations of the standard basis
vectors; for example (x, y, z) = x(1, 0, 0) + y(0, 1, 0) + z(0, 0, 1). However, under some circumstances one may wish to change the basis, that is, see the points of the vector space as linear
combinations of an alternative basis: (x, y, z) = x(1, 0, 1) + y(0, 1, 1) + (z − x − y)(0, 0, 1).
Second, when considering geometric objects, a symmetry is a transformation that appears
to leave the object unchanged; for example, rotating a square through 90◦ produces a result
indistinguishable from the original situation. Thus, finding symmetries involves transforming
an object (by rotating it, reflecting it, shifting it, scaling it, etc.) and considering to what
extent the result resembles the original. This kind of seeing-as becomes more interesting as
the objects become more complicated and partial alignments of geometric structure become
more frequent than total alignments (as happens with aperiodic tilings, for example). That
the latter variety of mathematical seeing-as could be relevant here is suggested by Lakoff
and Johnson: “Understanding a conversation as being an argument involves being able to
superimpose the multidimensional structure of part of the concept WAR upon the corresponding structure CONVERSATION.”47 Their METAPHOR IS SLIDE PROJECTION metametaphor developed
out of this approach.48 Gregory Bateson provides further inspiration to pursue this line of
thinking in his discussion of moiré phenomena (when two superimposed patterns generate
a third): “it becomes possible to investigate an unfamiliar pattern by combining it with
a known second pattern and inspecting the third pattern which they together generate.”49
Perhaps metaphor can be seen as involving a superimposition of function graphs?
A substantial mathematical theory exists that spectacularly combines these various ideas.
Fourier analysis involves seeing functions as sums of sine and cosine waves. That this is math46
Mappings between function spaces are often referred to as operators.
Metaphors We Live By 81.
48
Lakoff and Johnson, Metaphors We Live By 253–4.
49
Gregory Bateson, Mind and Nature (New York: E.P. Dutton, 1979), 79.
47
198
ematically robust depends on the fact that an orthonormal basis of the function space L2
can be made from sine and cosine waves.50 If the function under consideration is periodic, it
can be written as a (possibly infinite) Fourier series, that is, as a linear combination of the
wave basis vectors. However, a wider variety of square-integrable functions can be handled
by way of the Fourier transform which produces a continuous rather than discrete interpretation of the function in terms of waves.51 The Fourier transform has become an extremely
valuable tool for scientists and engineers because all empirical waveforms have compact support and are bounded and are therefore square integrable. Real-world applications such as
signal storage and compression, noise filtering, and pattern detection involve Fourier analysis. The Fourier transform is a mapping from a function to a function which involves a kind
of mathematical seeing-as; perhaps metaphor can be understood as a Fourier transform?
There are at least two significant problems with this idea. First, Fourier analysis always
involves a seeing-as in terms of sine and cosine functions, whereas metaphor must be able
to accommodate a much wider variety of source domains.52 Second, the result of applying a
Fourier transform to a waveform is not another waveform of the same variety but a frequency
spectrum; that is, Fourier transforms are usually understood to map functions defined on
a time domain to functions defined on a frequency domain. Applying Fourier analysis to a
continuous periodic function produces a discrete set of frequency coefficients; there is thus
no meaningful resemblance between the output of a Fourier transform and either its input
function or a sine wave. The output of a Fourier transform is not a function graph of the
right kind to be a concept, and therefore it cannot be the metaphor mapping.
50
Rajendra Bhatia, Fourier Series (Washington, DC: Mathematical Association of America, 2005), 10.
Orthogonality of vectors is a generalization of geometric perpendicularity defined in terms of an inner product.
A vector is normal if it has norm 1; a norm is a function that generalizes the notion of the length or magnitude
of a vector, and is also often defined in terms of an inner product. A basis, then, is orthonormal if all its
vectors have norm
n R1 and are omutually orthogonal. L2 is the function space of square-integrable functions;
2
that is, L2 = f : |f | < ∞ . Technically, the orthonormal basis of sines and cosines mentioned exists for
a version of L2 restricted to the interval [−π, π] (Bhatia 9).
R∞
51
The standard Fourier transform is defined by fˆ(t) = −∞ f (x)e−itx dx (Bhatia 11). Fourier analysis is
not confined to R; versions of the Fourier transform exist for functions defined on a wide range of spaces and
manifolds (Triebel 4).
52
The idea that all functions can be understood in terms of waves is analogous to the thought that mathematics provides a universal source domain, that everything in the world can be understood in terms of
mathematics. Given this observation, it is thus perhaps not surprising that Jean Baptiste Joseph Fourier
himself once said “Mathematics compares the most diverse phenomena and discovers the secret analogies
that unite them” (qtd. in du Sautoy).
199
All is not lost. A few modifications will rectify these difficulties and show that integral
transforms provide a useful way to understand metaphor. First, mathematicians have developed and studied a variety of integral transforms akin to the Fourier transform but based
upon different analyzing functions. Of particular note are the wavelet transforms, integral
transforms similar to the Fourier transform but based on a wide variety of functions. Each of
these “mother wavelet” functions possesses specific properties that make it useful for understanding various interesting features of a waveform’s structure, not only its periodicities.53
There are an infinite number of possible wavelets, and given a particular signal to analyze
and purpose for analyzing it, some of those wavelets will perform better (i.e., more efficiently
and accurately) than others.54 Indeed, it may be the case that different wavelets are better
suited to handling different parts of a waveform — the different movements of a symphony in
a single audio recording, for example. The idea that an integral transform could be derived
from any given source domain concept (perhaps through the creation of a custom wavelet
based on the source function) overcomes the first problem with conceiving metaphors as
transforms. The second problem can be resolved by composing the integral transform with
its inverse transform, a mapping which recreates (an approximation of) the original function
from its frequency spectrum.55 The result will be a new function that approximates the
analyzed function, but possibly with some aspects of the analyzing function incorporated.
Finally, an updated, refined version of the original Lakovian metametaphor has been reached:
METAPHOR IS A TRANSFORM-INVERSE TRANSFORM COMPOSITION MAPPING.
This
TRANSFORM
metametaphor has several promising features to recommend it. First,
insofar as it is an elaboration of the Lakovian metametaphor
CAL MAPPINGS,
METAPHORS ARE MATHEMATI-
it inherits the positive features of its ancestral mapping recounted earlier in
this chapter. In particular, the
TRANSFORM
metametaphor does a good job of capturing the
53
Barbara Burke Hubbard, The World According to Wavelets (Wellesley: AK Peters, 1998), 32–3. As even
the most basic explanation of wavelet analysis would be both relatively inaccessible and unnecessary for the
purposes of this metaphorical discussion, I refer the interested reader to the excellent resource cited here, a
refreshing diptych of mathematical precision and rigor juxtaposed with widely accessible sections discussing
the historical development and practical repercussions of wavelet analysis.
54
Hubbard 239.
55
A composition of a transform with its inverse is one way that noise filtering can be accomplished:
transform a waveform into its frequency spectrum (that is, analyze it), eliminate frequencies associated with
noise from the spectrum (usually by discarding coefficients smaller than some given threshold), then rebuild
the waveform from the modified frequency spectrum using the inverse transform(synthesize it).
200
asymmetry of metaphor: the roles of the analyzing function and the analyzed function in a
transform are significantly different, and this difference is exhibited in the output of the transform. There is also some reason to believe that the
TRANSFORM
metametaphor could cohere
with a neurological understanding of metaphor like the one Lakoff now espouses. The fact
that neuroscientists interpret neural oscillations — often called brain waves — as symptoms
of brain activity of various kinds seems coherent with the idea that concepts are waveforms.
Moreover, there is empirical evidence that our perceptive pathways perform Fourier analysis;
it thus seems reasonable that metaphor and other cognitive mechanisms could involve transforms as well.56 However, the main advantage of METAPHOR IS A TRANSFORM-INVERSE TRANSFORM
COMPOSITION MAPPING
is arguably that it can explain the creation of target domain entities,
since the inability of
METAPHORS ARE MATHEMATICAL MAPPINGS
Recall that many abstract concepts —
LOVE,
to do so led to its retirement.
for example — possess minimal inherent struc-
ture so inherit much of their structure via conceptual metaphors.57 Since a mathematical
mapping cannot add new elements to its codomain but only specifies connexions between two
preexisting sets, a metaphor cannot create target entities insofar as the target domain of the
metaphorical mapping is understood as its codomain. Under the proposed metametaphor,
however, a target concept waveform is transformed into a new waveform that is seen as a
modified version of the original concept; taking the difference of the two waveforms exposes
the structures that have been created or modified by the mapping. In particular, consider
a target which is either a waveform that has gaps in it or which is merely a discrete set of
points. Interpolation involves finding a curve that connects the given datapoints of the target,
replacing gappy data with a continuous function; the act of interpolating can be understood
as creating the entities that fill the gaps. While there are a variety of approaches to interpolation, wavelets provide one way of doing the job.58 Thus, the composition of transforms
provides a way of seeing a target concept in terms of another even if the target is sparse.
A waveform and its transform are two sides of the same coin, representations of the
same object that emphasize different aspects. This relationship provides an opportunity
56
See De Valois, De Valois, and Yund (1978) for details on Fourier analysis in visual perception. Wavelets
have also been hypothesized as playing a role in visual and auditory perception (Hubbard 69–70).
57
Lakoff and Johnson, Philosophy in the Flesh 70–1.
58
Yves Nievergelt, Wavelets Made Easy (Boston: Birkhäuser, 1999), 82–3.
201
to metaphorically connect CMT to other promising contemporary theories of concepts and
metaphor discussed in chapter 3. Jesse Prinz’s theory of concepts as proxytypes builds upon
Lawrence Barsalou’s claim that “concepts cannot be identified with the totality of category
knowledge stored in long-term memory.”59 Prinz views concepts as “mental representations
of categories that are or can be activated in working memory.”60 When a concept is required,
its instantiation in working memory is shaped by the context of activation. The
TRANSFORM
metametaphor provides a potentially useful way of understanding proxytypes. Knowledge
networks stored in long-term memory could be interpreted as frequency spectra. From time
to time, a circumstance arises — a perception, a memory, a thought — that requires representation, “highlighting” some components of the frequency spectrum.61 A proxytype would
then be activated in working memory by applying an inverse transform to the highlighted
spectrum components. Conversely, active thoughts and experiences could modify our dormant knowledge networks through an application of the transform. This is an appealing
view insofar as it mirrors the way media files on a computer are handled. A visual or sonic
waveform is transformed and stored as frequency coefficients; it is typically more efficient
to encode a waveform as its frequency spectrum than to represent it directly. When a user
wants to access the media in the file, the computer decodes the information by applying
the inverse transform to recreate the waveform. One question that needs to be addressed
is what inverse transform is being applied in the activation of a proxytype? Whereas the
source domain concept that defines the transform is specified in the case of metaphor, no
such indication is made in this case. One possible way around this problem is to posit the
existence of a standard, default conceptual transform; this would have the advantage of providing one way of differentiating between conceptual metaphor and other cognitive mappings.
These metaphorical elaborations on the nature of concepts may also provide resources for
conceptualizing some of Cornelia Müller’s important distinctions. A novel metaphor could
be interpreted as having an analyzing wavelet that is still under development, an entrenched
metaphor would have a well-established transform, and a dead metaphor’s transform would
59
Prinz 148.
Prinz 149; emphasis his.
61
This highlighting would probably involve a combination of interaction between frequency functions with
some thresholding.
60
202
no longer exist, the analyzing function having been lost. Additionally, the sleeping/waking
dimension of Müller’s theory could correspond with the amount of thresholding that occurs in
the composite mapping process and therefore with the amount of the analyzing function that
shows through in the result.62 These metaphorical sketches are a bit rough, but do provide
further support for the idea that CMT could be fruitfully integrated with other promising
theories.
Mathematical metametaphors provide a useful way of understanding metaphor. The
function-space and transform-operator metaphors introduced in this chapter not only rejuvenate and improve upon Lakoff and Johnson’s original metametaphorical understanding of
conceptual metaphor, but also provide potential connexions to both their own contemporary
neurological understanding as well as some noteworthy alternative philosophical theories of
concepts and metaphor. The metaphorical work that has been done here is promising, but
also very preliminary and rough; the implications of this approach are far from fully realized. In particular, very little has been said about the disanalogous aspects of the TRANSFORM
metametaphor — and there are likely to be many. Readers interested in developing this
metametaphor further may wish to first focus on the
CONCEPTS ARE FUNCTIONS
part of the
metaphor; in particular, very little was said about how to understand intercategorial structure within the conceptual function space. Readers interested in the possibility of developing this approach less metaphorically and more rigorously should seek out the disanalogies
and consider whether they can be circumvented. There are certainly other mathematical
metametaphors that could be considered; the specific approach taken here is primarily a
function of my specific mathematical background, which featured undergraduate research on
wavelet analysis and graduate work studying aperiodic tilings using mathematical diffraction
(a phenomenon understood in terms of Fourier transforms). Finally, to those readers who
have somehow made it to the end of the chapter despite finding the approach taken in this
chapter bewildering and distasteful, I encourage you to consider this writing not as sloppy
mathematics but more akin to strange (and probably bad) poetry: it does aim at the truth,
but it uses a shotgun.
62
Müller 11.
203
Chapter 6
Conclusion
Hofstadter’s Law: It always takes longer than you think it will take,
even when you take into account Hofstadter’s Law.1
— Douglas Hofstadter
Endings are often difficult, and such is the case here. This project has been an integral
part of my existence for nearly a third of my life, and, as it comes to a close, I feel as though
I am losing an important part of myself. Furthermore, the project feels incomplete insofar
as that there is so much more to be said about mathematics and metaphor. However, it
is more healthy — and more apt — to conceive of the denouement of my dissertation as
metamorphic rather than injurious or abortive. To paraphrase a pearl of wisdom imparted
to me by several individuals over the last few years, a doctorate is meant to be the beginning
of a scholastic career, not the culmination of one.2 That is, though the current project must
end without being fully definitive and conclusive, research on this topic can (and should)
continue for years to come. Thus, to conclude, I briefly review what this dissertation has
accomplished, and indicate some prospects for future research and application.
It seems impossible to provide a comprehensive survey of metaphor scholarship in a
single volume. As Wayne Booth noted in 1977, “[e]xplicit discussions of something called
metaphor have multiplied astronomically in the past fifty years. . . students of metaphor have
positively pullulated,” an observation which remains relevant over thirty years later.3 The
best anyone can hope to do is summarize the important highlights. Given that other authors
(such as Johnson and Leezenberg) have provided good synopses of metaphor elsewhere that
1
Hofstadter, I am a Strange Loop (New York: Basic Books, 2007), xiv.
It can be somewhat difficult to remember this within the current economic climate, where academic
positions are scarce and the competition is fierce.
3
Wayne C. Booth, “Metaphor as Rhetoric: The Problem of Evaluation,” On Metaphor, Ed. Sheldon Sacks
(Chicago: U Chicago P, 1978), 47.
2
204
I have made reference to, what does my overview contribute? First, unlike the synopses
of other authors, mine is not elsewhere: it provides the reader with a convenient and easily
accessible resource for obtaining the understanding of metaphor necessary for comprehending
the rest of the dissertation. And second, while there is necessarily considerable overlap
between the synopsis provided here and those of other authors, there are also some noteworthy
differences. In particular, my précis of metaphor in chapters 2 and 3 emphasizes the history
and development of conceptual metaphor theories. Several scholars — both historical and
contemporary — connected to that tradition that are downplayed or overlooked in other
synopses receive attention here, including Vico, Nietzsche, Ricoeur, Eco, and Müller. Though
the contribution to the overall project is arguably minimal, it is noteworthy that I have
never encountered discussion of Nicomachus in any other writings on metaphor. One obvious
direction for future research would be to expand and improve upon the literature review of
metaphor scholarship started here.
This dissertation argues that metaphor is conceptual in nature, but does not offer a novel
theory of conceptual metaphor. Instead, Lakoff’s CMT is defended as a plausible — albeit
flawed — version of a theory of metaphor as conceptual. While many others have defended
the Lakovian position against critics, the defense provided here differs from the others in
a few key ways. First, supporters and critics of CMT alike tend to focus their attention
on one or two major works in the Lakovian oeuvre. The synopsis of CMT I provide in
chapter 3 is more comprehensive than most, making explicit reference to more than a dozen
books and papers published by Lakoff, Johnson, and their collaborators. This thorough
approach overcomes criticisms dependent upon either an oversimplified understanding of
CMT or the equivocal importation of traditional linguistic assumptions. Second, my defense
of conceptual metaphor is structured by
CONCEPT,
categorizing the various strengths and
weaknesses of CMT. This approach emphasizes the fact that any theory of metaphor as
conceptual must be affiliated with an account of concepts, and it exposes commonalities
between CMT and other conceptual approaches that future authors might exploit to create
a novel robust hybrid. Ultimately, though I think that there is much to be said in favour
of the Lakovian approach, I am not wholeheartedly committed to the specifics of CMT but
only to a conceptual approach to metaphor in general. Given the centrality of the notion
205
CONCEPT
to philosophy and cognitive science, and given the current amount of disagreement
on the topic, it seems there will be a need for scholarship in this area for years to come.
The overview of metaphor provided in chapters 2 and 3 is a necessary and interesting
component of my project, but is ultimately peripheral: the heart of this dissertation is the
argument that metaphor plays a constitutive role in mathematical practice. This general
approach is motivated in part by the observation that the traditionally popular philosophical
theories of mathematics each capture some important aspect of mathematics, but also possess
fatal flaws that make them untenable. Adopting a framework that emphasizes conceptual
metaphor provides a mechanism that allows for a partial reconciliation of these apparently
incompatible theories of mathematics; chapter 4 begins this reconciliation process. Chapter
4 also considers several specific contemporary theories of mathematics amenable to the idea
of constitutive mathematical metaphor, and claims that coherences exist between them that
could allow them to collaboratively support each other rather than competing. In particular,
the suggestion that Lakovian embodied mathematics and Yablonian figuralism are compatible
seems obvious yet has not been discussed in the literature before now. Thus, like chapter 3,
chapter 4 does not provide a novel theory but rather argues in favour of the development of a
complex hybrid theory of mathematics with conceptual metaphor at its core that incorporates
the strengths of a variety of theories usually viewed as rivals. Some important preliminary
groundwork has been provided here, but the realization of such a hybrid theory will require
significantly more work; this is an obvious direction for future research.
Chapter 5 argues that, even if the arguments of chapter 4 are airtight, it is nonetheless
useful and non-question-begging to understand metaphor in terms of mathematics. Rather
than looking to develop a rigorous computational model of metaphor, I instead sketch a
novel metametaphor mapping the mathematical source domain of
the target domain
METAPHORS.
WAVELET TRANSFORMS
to
The brevity and speculative nature of this chapter presents
many opportunities for future research. First, the paltry history of computational linguistics
provided could be significantly expanded and the state of the art of cognitive and linguistic
modeling could be surveyed. Second, the
TRANSFORM
metametaphor sketched in chapter 5
could be further developed and refined. Third, additional novel mathematical metametaphors
could be developed that help enrich our understanding of metaphor while simultaneously
206
lending support to this somewhat unorthodox and taboo use of mathematical concepts. And,
fourth, the possibility of adapting the metametaphor given in chapter 5 into a rigorous
mathematical model of metaphor could be seriously entertained. Such an inquiry might lead
to the development of such a model, or to an argument explaining why such a model is not
feasible (or outright impossible). Either of these outcomes would be rewarding.
The repercussions of my research are primarily theoretical rather than practical. Mathematics is clearly a highly productive and successful endeavour as it currently stands, and
my research aims to describe and explain mathematical practices, not to alter them. The
impact of my research, if any, will therefore be felt in the philosophy of mathematics, not
within mathematics itself. If there is one area where I might hope my research could be
applied, it is in mathematics education. Whereas mathematics research is highly successful
as it stands, many people learn to dislike and avoid mathematics. While not every person
needs to be a professional mathematician, the combination of increasing the average level of
mathematical competency while simultaneously decreasing the amount of animosity towards
mathematics could only be beneficial in this increasingly scientific age. Insofar as metaphor
plays a constitutive role in mathematics, and insofar as metaphor is one of the most effective mechanisms humans have for comprehending the unknown, it seems that strategically
augmenting the existing mathematics curriculum with metaphorical explanations may be an
effective way of improving mathematics education. Though it clearly does not constitute a
controlled scientific study, over the last decade I have noticed an improvement in the uptake of my mathematics pupils that correlates with my becoming more mindful of using apt
metaphors in conjunction with rigorous definitions and proofs. I am therefore convinced that
such an approach has merit, though it is clear that more rigorous evidence of its efficacy is
desirable.
207
References
Aristotle. Poetics. Trans. I. Bywater. The Complete Works of Aristotle. Vol. 2. Ed. Jonathan
Barnes. Princeton: Princeton UP, 1984. 2316–40. Print.
–––. Rhetoric. Trans. J.H. Freese. Aristotle in 23 Volumes. Vol. 22. Cambridge: Harvard UP,
1926. Perseus Digital Library. Tufts U. Web. 22 Feb. 2009.
–––. Rhetoric. Trans. W. Rhys Roberts. The Complete Works of Aristotle. Vol. 2. Ed. Jonathan
Barnes. Princeton: Princeton UP, 1984. 2152–269. Print.
Athenaeus. The Learned Banqueters. Vol. 3. Ed. and trans. Douglas Olson. Cambridge: Harvard UP, 2008. Print.
Austin, John L. How to Do Things With Words. Ed. J.O. Urmson and Marina Sbisà. 2nd ed.
Oxford: Clarendon, 1975. Print.
Balaguer, Mark. “Fictionalism in the Philosophy of Mathematics.” Stanford Encyclopedia of
Philosophy. 16 Sept. 2011. Web. 30 Jul. 2012.
Barrow, John D. Pi in the Sky: Counting, Thinking, and Being. Oxford: Clarendon, 1992.
Print.
Barsalou, Lawrence W., et al. “Concepts and Meaning.” Chicago Linguistics Society 29:
Papers from the parasession on conceptual representations. Ed. K. Beals et al. Chicago:
Chicago Linguistics Society, 1993. 1–41. Lawrence Barsalou Website. Emory University.
Web. 26 Nov. 2011.
Bateson, Gregory. Mind and Nature. New York: E.P. Dutton, 1979. Print.
Benacerraf, Paul. “Mathematical Truth.” The Journal of Philosophy 70.19 (1973): 661–79.
JSTOR. Web. 17 May 2012.
Benacerraf, Paul and Hilary Putnam, eds. Philosophy of Mathematics: Selected Readings.
Englewood Cliffs: Prentice-Hall, 1964. Print.
Bhatia, Rajendra. Fourier Series. Washington, DC: Mathematical Association of America,
2005. Print.
Black, Max. “Metaphor.” Proc. of Aristotelian Society 55 (1954–55): 273–94. Reprinted in
Johnson, Philosophical Perspectives on Metaphor 63–82. Print.
–––. Models and Metaphors. Ithaca: Cornell UP, 1962. Print.
–––. “More About Metaphor.” 19–41. Ortony, Metaphor and Thought 19–41. Print.
–––. “Review of Metaphors We Live By.” J. of Aesthetics and Art Criticism 40.2 (1981):
208–10. JSTOR. Web. 8 May 2009.
Bloor, David. Knowledge and Social Imagery. 2nd ed. Chicago: U of Chicago P, 1991. Print.
Bolzano, Bernard. Contributions to a Better-Grounded Presentation of Mathematics. Prague:
Kaspar Widtmann, 1810. Found in The Mathematical Works of Bernard Bolzano. Ed. and
Trans. Steve Russ. New York: Oxford UP, 2004. 83–137. MyiLibrary. Web. 31 Jan. 2012.
Booth, Wayne C. “Metaphor as Rhetoric: The Problem of Evaluation.” On Metaphor. Ed.
Sheldon Sacks. Chicago: U Chicago P, 1979. 47–70. Print.
208
Boroditsky, Lera. “Metaphoric structuring: understanding time through spatial metaphors.”
Cognition 75.1 (2000): 1–28. ScienceDirect. Web. 2 Oct. 2013.
Brouwer, L.E.J. Brouwer’s Cambridge Lectures on Intuitionism. Ed. D. van Dalen. Cambridge: Cambridge UP, 1981. Print.
Brown, James Robert. Philosophy of Mathematics. 2nd ed. New York: Routledge, 2008. Print.
Burgess, John P. “Mathematics and Bleak House.” Philosophia Mathematica 12.1 (2004):
18–36. EBSCOhost. Web. 15 Apr. 2013.
Camp, Elisabeth. “Two Varieties of Literary Imagination: Metaphor, Fiction, and Thought
Experiments.” Midwest Studies in Philosophy. Ed. Peter A. French, Howard K. Wettstein,
and Ernie Lepore. Vol. 33. Boston: Wiley-Blackwell, 2009. 107–30. Print.
Camp, Elisabeth and Marga Reimer. “Metaphor.” Handbook of Philosophy of Language. Ed.
Lepore and Smith. Oxford: Oxford UP, 2006. 845–63. Print.
Carazo, Pau, et al. “Quantity discrimination in Tenebrio molitor: evidence of numerosity
discrimination in an invertebrate?” Animal Cognition 12.3 (2009): 463–70. SpringerLink.
1 Jan. 2009. Web. 11 Aug. 2012.
Carlyle, Thomas. Past and Present. Ed. Chris R. Vanden Bossche. Berkeley: U of California
P, 2005. Print.
Chalmers, David, Robert French, and Douglas Hofstadter. “High-level Perception, Representation, and Analogy: A Critique of Artificial-intelligence Methodology.” Fluid Concepts
and Creative Analogies. New York: Basic Books, 1995. 169–93. Print.
“concept, n.” OED Online. Mar. 2014. Oxford UP. Web. 25 Apr. 2014.
“concipiō.” Oxford Latin Dictionary. 1968. Print.
Costelloe, Timothy. “Giambattista Vico.” Stanford Encyclopedia of Philosophy. 15 Jul. 2008.
Web. 11 Mar. 2009.
Dabbs, James M. and Joseph P. Moorer. “Core Body Temperature and Social Arousal.”
Personality and Social Psychology Bulletin 1.3 (1975): 517–20. SAGE Journals Online.
Web. 4 Jun. 2010.
Dacke, Marie and Madyam V. Srinivasan. “Evidence for counting in insects.” Animal Cognition 11.4 (2008): 683–89. SpringerLink. 27 May 2008. Web. 22 Jun. 2011.
Danesi, Marcel. “A conceptual metaphor framework for the teaching of mathematics.” Studies
in Philosophy and Education 26.3 (2007): 225–36. EBSCOhost. Web. 8 Mar. 2009.
Davidson, Donald. “A Nice Derangement of Epitaphs.” Truth, Language, and History. New
York: Oxford UP, 2005. 89–107. Print.
–––. “Locating Literary Language.” Truth, Language, and History. New York: Oxford UP,
2005. 167–81. Print.
–––. “Reply to Oliver Scholz.” Reflecting Davidson: Donald Davidson Responding to an
International Forum of Philosophers. Ed. Rolf Stoecker. New York: de Gruyter, 1993.
172–73. Print.
–––. “Truth and Meaning.” Synthese 17 (1967): 304–23. Reprinted in Martinich, The Philosophy of Language 98–109. Print.
–––. “What Metaphors Mean.” On Metaphor. Ed. Sheldon Sacks. Chicago: U of Chicago P,
1979. 29–46. Reprinted in Martinich, The Philosophy of Language 435–46. Print.
Dawson, John W. Logical Dilemmas: The Life and Work of Kurt Gödel. Wellesley, MA: AK
Peters, 1997. Print.
209
de Bruijn, N.G. “Algebraic theory of Penrose’s non-periodic tilings of the plane I.” Indagationes Mathematicae 84.1 (1981): 39–52. ScienceDirect. Web. 23 Oct. 2013.
Dehaene, Stanislas. The Number Sense: How the Mind Creates Mathematics. 2nd ed. New
York: Oxford UP, 2012. Print.
Descartes, René. Discourse on Method. Trans. Donald Cress. 3rd ed. Indianapolis: Hackett,
1998. Selections reprinted in René Descartes: Philosophical Essays and Correspondence,
Ed. Roger Ariew, (Indianapolis: Hackett, 2000), 46–82. Print.
Detlefsen, Michael. “On an alleged refutation of Hilbert’s program using Gödel’s first incompleteness theorem.” Journal of Philosophical Logic 19.4 (1990): 343–77. JSTOR. Web. 7
May 2014.
Deutscher, Guy. The Unfolding of Language. New York: Metropolitan, 2005. Print.
De Valois, Karen, Russell De Valois, and E. William Yund. “Responses of Striate Cortex
Cells to Grating and Checkerboard Patterns.” Journal of Physiology 291 (1979): 483–
505. PubMed. Web. 3 May 2010.
Devlin, Keith. Mathematics: The Science of Patterns. New York: Scientific American Library,
1997. Print.
du Sautoy, Marcus. “Joseph Fourier.” A Brief History of Mathematics. BBC Radio 4. Podcast
audio. 16 Jun. 2010. MP3. 2 Feb. 2012.
Eco, Umberto. “Metaphor, Dictionary, and Encyclopedia.” New Literary History 15.2 (1984):
255–71. JSTOR. Web. 12 Jan. 2012.
–––. Semiotics and the Philosophy of Language. Bloomington: Indiana UP, 1984. Print.
Ernest, Paul. Social Constructivism as a Philosophy of Mathematics. Albany: SUNY Press,
1998. Print.
Falkenhainer, Brian, Kenneth D. Forbus, and Dedre Gentner. “The Structure-Mapping Engine: Algorithm and Examples.” Artificial Intelligence 41 (1989): 1–63. Dedre Gentner’s
Web Page. Northwestern University. Web. 13 Aug. 2013.
Fauconnier, Gilles. Mappings in Thought and Language. New York: Cambridge UP, 1997.
Print.
Fauconnier, Gilles and Mark Turner. The Way We Think: Conceptual Blending and the
Mind’s Hidden Complexities. New York: Basic Books, 2002. Print.
Field, Hartry. Realism, Mathematics and Modality. Oxford: Blackwell, 1989. Print.
–––. Science Without Numbers. Princeton: Princeton UP, 1980. Print.
Fodor, Jerry A. Concepts: Where Cognitive Science Went Wrong. New York: Oxford UP,
1998. Print.
Freeman, Kathleen. Ancilla to the Pre-Socratic Philosophers. Cambridge: Harvard UP, 1948.
Print.
Frege, Gottlob. The Foundations of Arithmetic. Trans. J.L. Austin. 2nd ed. New York: Harper
and Brothers, 1953. Print.
Gallese, Vittorio and George Lakoff. “The Brain’s Concepts.” Cognitive Neuropsychology 22.3
(2005): 455–79. EBSCOhost. Web. 15 Mar. 2010.
Gentner, Dedre. “Structure Mapping: A Theoretical Framework for Analogy.” Cognitive Science 7 (1983): 155–70. Dedre Gentner’s Web Page. Northwestern University. Web. 17 Feb.
2010.
210
Gentner, Dedre, et al. “Metaphor Is Like Analogy.” The Analogical Mind: Perspectives from
Cognitive Science. Ed. Dedre Gentner, Keith J. Holyoak, and Boicho N. Kokinov. Cambridge: MIT Press, 2001. 199–253. Print.
Gibbs, Raymond W. “Evaluating Conceptual Metaphor Theory.” Discourse Processes 48.8
(2011): 529–62. Taylor and Francis Online. Web. 1 Oct. 2013.
–––, ed. The Cambridge Handbook of Metaphor and Thought. New York: Cambridge UP,
2008. Print.
–––. The Poetics of Mind. New York: Cambridge UP, 1994. Print.
–––. “Why many concepts are metaphorical.” Cognition 61 (1996): 309–19. ScienceDirect.
Web. 23 Apr. 2006.
Gold, Bonnie. “Read This! The MAA Online book review column: Where Mathematics Comes
From: How the Embodied Mind Brings Mathematics Into Being.” MathDL. The Mathematical Assoc. of America, 3 Feb. 2001. Web. 5 May 2009.
Goldin, Gerald A. “Review: Counting on the Metaphorical.” Nature 413.6851 (2001): 18–19.
EBSCOhost. Web. 22 Aug. 2007.
Gómez-Laplaza, Luis M. and Robert Gerlai. “Spontaneous discrimination of small quantities:
shoaling preferences in angelfish (Pterophyllum scalare).” Animal Cognition 14.4 (2011):
565–74. SpringerLink. 16 Mar. 2011. Web. 22 Jun. 2011.
Grady, Joseph. “Foundations of Meaning: Primary Metaphors and Primary Scenes.” Diss.
Berkeley, 1997. Print.
Grady, Joseph E., Todd Oakley, and Seana Coulson. “Blending and metaphor.” Metaphor
in Cognitive Linguistics. Ed. Raymond W. Gibbs and Gerard J. Steen. Amsterdam and
Philadelphia: John Benjamins, 1999. 101–24. Print.
Grattan-Guiness, Ivor. The Norton History of the Mathematical Sciences. New York: W.W.
Norton and Company, 1997. Print.
Grice, H.P. “Logic and Conversation.” Syntax and Semantics. Ed. Peter Cole and Jerry
L. Morgan. Vol. 3. New York: Academic P, 1975. 41–58. Reprinted in Martinich, The
Philosophy of Language 165–175. Print.
–––. “Meaning.” Philosophical Review 66 (1957): 377–88. Reprinted in Martinich, The Philosophy of Language 92–97. Print.
Guhe, Markus, Alan Smaill, and Alison Pease. “A Formal Cognitive Model of Mathematical
Metaphors.” KI 2009: Advances in Artificial Intelligence. 32nd Annual German Conference on AI, Paderborn, Germany, September 15-18,2009. Ed. B. Mertsching, M Hund,
and Z Aziz. Berlin: Springer, 2009. 323–30. Alison Pease’s Web Page. University of Edinburgh School of Informatics. Web. 4 Feb. 2013.
Hilbert, David. David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917–
1933. Ed. William Ewald and Wifried Sieg. Heidelberg: Springer-Verlag, 2013. Print.
–––. “Mathematical Problems.” Bulletin of the American Mathematical Society 8.10 (1902):
437–79. Project Euclid. Web. 7 May 2014.
–––. “On the Infinite.” Trans. Erna Putnam and Gerald J. Massey. Benacerraf and Putnam.
134–51. Originally published in Mathematische Annalen 95 (1925): 161–90. Print.
Hinks, D.A.G. “Tisias and Corax and the Invention of Rhetoric.” Classical Quarterly 34.1
(1940): 61–9. JSTOR. Web. 25 Apr. 2014.
Hobbes, Thomas. Leviathan. Ed. Edwin Curley. Indianapolis: Hackett, 1994. Print.
211
Hoffman, Sarah. “Mathematics as Make-Believe: A Constructive Empiricist Account.” Diss.
U of Alberta, 1999. Print.
–––. “Yablo’s Figuralist Account of Mathematics.” (2012): 1–20. Unpublished Article.
Hofstadter, Douglas R. “Analogies and Metaphors to Explain Gödel’s Theorem.” The TwoYear College Mathematics Journal 13.2 (1982): 98–114. JSTOR. Web. 18 Feb. 2013.
–––. “Epilogue: Analogy as the Core of Cognition.” The Analogical Mind: Perspectives from
Cognitive Science. Ed. Dedre Gentner, Keith J. Holyoak, and Boicho N. Kokinov. Cambridge: MIT Press, 2001. 499–538. Print.
–––. I am a Strange Loop. New York: Basic Books, 2007. Print.
Hofstadter, Douglas R. and the Fluid Analogies Research Group. Fluid Concepts and Creative
Analogies. New York: Basic Books, 1995. Print.
Holyoak, Keith J. “Overview of Research Career.” Keith Holyoak’s Web Page. UCLA Reasoning Lab. 31 Jul. 2012. Web. 18 Aug. 2013.
Holyoak, Keith J., Dedre Gentner, and Boicho N. Kokinov. “Introduction: The Place of
Analogy in Cognition.” The Analogical Mind: Perspectives from Cognitive Science. Ed.
Dedre Gentner, Keith J. Holyoak, and Boicho N. Kokinov. Cambridge: MIT Press, 2001.
1–20. Print.
Holyoak, Keith J. and Paul Thagard. “Analogical Mapping by Constraint Satisfaction.”
Cognitive Science 13 (1989): 295–355. Keith Holyoak’s Web Page. UCLA Reasoning Lab.
31 Jul. 2012. Web. 18 Aug. 2013.
Horsten, Leon. “Philosophy of Mathematics.” Stanford Encyclopedia of Philosophy. 2 May
2012. Web. 7 May 2012.
Hubbard, Barbara Burke. The World According to Wavelets: The Story of a Mathematical
Technique in the Making. 2nd ed. Wellesley: AK, 1998. Print.
“Inside Google Translate.” Google Translate. Google, n.d. Web. 28 Jan. 2014.
Islamic Art and Geometric Design. New York: Museum of Modern Art, 2004. Google Books.
Web. 8 July 2013.
Isocrates. Cyprian Orations. Ed. Edward Forster. Oxford: Clarendon, 1912. Internet Archive.
Web. 25 Apr. 2014.
Jackendoff, Ray and David Aaron. “Review of More Than Cool Reason.” Language 67.2
(1991): 320–38. JSTOR. Web. 8 May 2009.
Johnson, Christopher. “Metaphor vs. conflation in the acquisition of polysemy: The case of
see.” Cultural, Psychological, and Typological Issues in Cognitive Linguistics. Ed. Masako
Hiraga, Chris Sinha, and Sherman Wilcox. Philadelphia: John Benjamins, 1999. 155–70.
Print.
Johnson, Mark. “Introduction: Metaphor in the Philosophical Tradition.”. M. Johnson, Philosophical Perspectives on Metaphor 3–47. Print.
–––, ed. Philosophical Perspectives on Metaphor. Minneapolis: U of Minnesota P, 1981. Print.
–––. The Body in the Mind. Chicago: U of Chicago P, 1986. Print.
Johnson, Mark and George Lakoff. “Why cognitive linguistics requires embodied realism.”
Cognitive Linguistics 13.3 (2002): 245–63. Gale Cengage. Web. 7 Nov. 2011.
Kirby, John T. “Aristotle on Metaphor.” American Journal of Philology 118 (1997): 517–54.
JSTOR. Web. 6 Jan. 2008.
Kitcher, Philip. The Nature of Mathematical Knowledge. New York: Oxford UP, 1984. Print.
212
Kline, Morris. Mathematical thought from ancient to modern times. New York: Oxford UP,
1972. Print.
Kövecses, Zoltán. Metaphor: A Practical Introduction. New York: Oxford UP, 2002. Print.
Kusch, Martin. “Psychologism.” Stanford Encyclopedia of Philosophy. 7 Nov. 2011. Web. 14
Apr. 2012.
Lakatos, Imre. Proofs and Refutations. Ed. John Worrall and Elie Zahar. New York: Cambridge UP, 1976. Print.
Lakoff, George. “The Contemporary Theory of Metaphor.” 202–51. Ortony, Metaphor and
Thought 202–51. Print.
–––. “The Death of Dead Metaphor.” Metaphor and Symbolic Activity 2.2 (1987): 143–7.
Informaworld. Web. 23 May 2009.
–––. “The Meanings of Literal.” Metaphor and Symbolic Activity 1.4 (1986): 291–6. EBSCOhost. Web. 20 Nov. 2011.
–––. “The Neural Theory of Metaphor.” 17–38. Gibbs, The Cambridge Handbook of Metaphor
and Thought 17–38. Print.
–––. Women, Fire, and Dangerous Things. Chicago: U of Chicago P, 1987. Print.
Lakoff, George and Mark Johnson. Metaphors We Live By. 1980. With a new afterword.
Chicago: U of Chicago P, 2003. Print.
–––. Philosophy in the Flesh. New York: Basic Books, 1999. Print.
Lakoff, George and Rafael E. Núñez. “Reply to Bonnie Gold’s review of ‘Where Mathematics
Comes From: How the Embodied Mind Brings Mathematics into Being’.” MathDL. The
Mathematical Assoc. of America, 3 Feb. 2001. Web. 5 May 2009.
–––. “The Metaphorical Structure of Mathematics: Sketching Out Cognitive Foundations
for a Mind-Based Mathematics.” Mathematical Reasoning: Analogies, Metaphors, and
Images. Ed. L. English. Hillsdale, NJ: Erlbaum, 1997. 21–89. Print.
–––. Where Mathematics Comes From. New York: Basic Books, 2000. Print.
Lakoff, George and Mark Turner. More Than Cool Reason. Chicago: U of Chicago P, 1989.
Print.
Langacker, Ronald W. “An Introduction to Cognitive Grammar.” Cognitive Science 10.1
(1986): 1–40. Elsevier ScienceDirect. Web. 29 Oct. 2011.
Laurence, Stephen and Eric Margolis. “Concepts and Cognitive Science.” Concepts: Core
Readings. Ed. Eric Margolis and Stephen Laurence. Cambridge: MIT Press, 1999. 3–81.
Print.
Leezenberg, Michiel. Contexts of Metaphor. Amsterdam: Elsevier, 2001. Print.
Leibniz, Gottfried. “On the Universal Science: Characteristic (Parts XIV and XV).” Monadology and Other Philosophical Essays. Ed. and trans. Paul Schrecker and Anne Martin
Schrecker. New York: Bobbs-Merrill, 1965. 11–21. Print.
Liddell, George Henry and Robert Scott. A Greek-English Lexicon. 9th ed. Oxford: Clarendon,
1940. Perseus Digital Library. Tufts U. Web. 4 Feb. 2009.
Linnebo, Øystein. “Platonism in the Philosophy of Mathematics.” Stanford Encyclopedia of
Philosophy. 25 Jul. 2011. Web. 2 May 2012.
“literal, adj. and n.” OED Online. Dec. 2011. Oxford UP. Web. 21 Feb. 2012.
Locke, John. An Essay Concerning Human Understanding. Ed. Peter H. Nidditch. Oxford:
Clarendon, 1975. Print.
213
Lu, Hongjing, Dawn Chen, and Keith J. Holyoak. “Bayesian Analogy with Relational Transformations.” Psychological Review 119.3 (2012): 617–48. Keith Holyoak’s Web Page. UCLA
Reasoning Lab. Web. 18 Aug. 2013.
Mac Cormac, Earl R. A Cognitive Theory of Metaphor. Cambridge: Bradford, 1985. Print.
Mac Lane, Saunders. Mathematics: Form and Function. New York: Springer-Verlag, 1986.
Print.
Manin, Yuri Ivanovich. “Mathematics as Metaphor.” Proceedings of the International Congress
of Mathematicians, August 21–29, 1990. Ed. Ichiro Satake. Vol. 2. Tokyo: Springer-Verlag,
1991. 1665–72. mathunion.org. Web. 27 Jan. 2013.
Martinich, A.P. “A Theory for Metaphor.” Journal of Literary Semantics 13 (1984): 35–56.
Reprinted in Martinich, The Philosophy of Language 447–458. Print.
–––. “A Theory of Fiction.” Philosophy and Literature 25 (2001): 96–112. Project Muse.
Johns Hopkins UP. Web. 5 Oct. 2006.
–––, ed. The Philosophy of Language. 4th ed. New York: Oxford UP, 2001. Print.
“mathematic, n. and adj.” The Oxford English Dictionary. 3rd ed. 2001. OED Online. Oxford
UP. Web. 17 May 2012.
McCall, Marsh H. Ancient Rhetorical Theories of Simile and Comparison. Cambridge: Harvard UP, 1969. Print.
McGlone, Matthew S. “What is the explanatory value of a conceptual metaphor?” Language
and Communication 27.2 (2007): 109–26. Elsevier ScienceDirect. Web. 29 Sept. 2008.
“metaphor, n.” OED Online. June 2008. Oxford UP. Web. 2 Feb. 2009.
“metonymy, n.” OED Online. Sept. 2011. Oxford UP. Web. 22 Sept. 2011.
Müller, Cornelia. Metaphors Dead and Alive, Sleeping and Waking. Chicago: U of Chicago
P, 2008. Print.
Murphy, Gregory L. “On metaphoric representation.” Cognition 60 (1996): 173–204. ScienceDirect. Web. 27 Sept. 2008.
–––. The Big Book of Concepts. Cambridge: MIT Press, 2002. Print.
Narayanan, Srinivas. “Embodiment in Language Understanding: Sensory-Motor Representations for Metaphoric Reasoning About Event Descriptions.” Diss. Berkeley, 1997. Print.
Nietzsche, Friedrich. “On Truth and Lying in an Extra-Moral Sense.” Friedrich Nietzsche
on Rhetoric and Language. Ed. Sander L. Gilman. Trans. David J. Parent. New York:
Oxford UP, 1989. 246–57. Print.
Nievergelt, Yves. Wavelets Made Easy. Boston: Birkhäuser, 1999. Print.
Núñez, Rafael. “Conceptual Metaphor, Human Cognition, and the Nature of Mathematics.”
Gibbs, The Cambridge Handbook of Metaphor and Thought 339–62. Print.
–––. “Numbers and arithmetic: neither hard-wired nor out there.” Biological Theory 4.1
(2009): 68–83. Núñez’s personal website at UCSD. Web. 9 Oct 2012.
Núñez, Rafael, Kensey Cooperrider, and Jürg Wassmann. “Number concepts without number
lines in an indigenous group of Papua New Guinea.” PLoS ONE 7.4 (2012): 1–8. Web. 9
Oct. 2012.
O’Connor, J.J. and E.F. Robertson. “Max Black.” MacTutor History of Mathematics Archive.
U of St. Andrews, Scotland., Mar. 2001. Web. 5 Mar. 2009.
Ogden, C.K. and I.A. Richards. The Meaning of Meaning. 8th ed. New York: Harcourt, Brace,
and Co., 1923. Print.
214
Ortony, Andrew. “Metaphor: A multidisciplinary problem.” Metaphor and Thought. Ed. Andrew Ortony. 1st ed. Cambridge: Cambridge UP, 1979. 1–18. Print.
–––, ed. Metaphor and Thought. 2nd ed. New York: Cambridge UP, 1993. Print.
Parsons, Glenn G. and James Robert Brown. “Platonism, Metaphor, and Mathematics.”
Dialogue 43.1 (2004): 47–66. Print.
Peacock, Thomas Love. “The Four Ages of Poetry.” Peacock’s Four Ages of Poetry, Shelley’s Defence of Poetry, Browning’s Essay on Shelley. Ed. H.F.B Brett-Smith. Oxford:
Blackwell, 1929. 3–19. Print.
Pimm, David. “Mathematical Metaphor.” For the Learning of Mathematics 8.1 (1988): 30–4.
JSTOR. Web. 29 Jan. 2013.
–––. “Metaphor and Analogy in Mathematics.” For the Learning of Mathematics 1.3 (1981):
47–50. JSTOR. Web. 27 Jan. 2013.
Plato. Meno. Trans. Benjamin Jowett. The Internet Classics Archive. Web Atomic and MIT.
Web. 14 Apr. 2014.
–––. Phaedrus. Trans. Harold N. Fowler. Plato in Twelve Volumes. Vol. 9. Cambridge: Harvard UP, 1925. Perseus Digital Library. Tufts U. Web. 18 Feb. 2009.
–––. Republic. Trans. Benjamin Jowett. The Internet Classics Archive. Web Atomic and
MIT, 7 Jan. 2009. Web. 6 Feb. 2009.
–––. Symposium. Trans. Benjamin Jowett. The Internet Classics Archive. Web Atomic and
MIT. Web. 8 Sept. 2009.
–––. Timaeus. Trans. Benjamin Jowett. The Internet Classics Archive. Web Atomic and
MIT. Web. 7 Feb. 2009.
Poincáre, Henri. “The Future of Mathematics.” The Monist 20.1 (1910): 76–92. JSTOR.
Web. 26 Feb. 2012.
Pólya, George. Induction and Analogy in Mathematics. Princeton: Princeton UP, 1954. Print.
Postnikoff, Derek. “Mathematical metaphor: Linguistic or conceptual?” 18th Novembertagung
on the History, Philosophy, and Didactics of Mathematics: Mathematical Practice and
Development throughout History. Ed. Ingo Witzke. Berlin: Logos Verlag, 2008. 3–14. Print.
Presmeg, Norma. “Review: Mathematical Idea Analysis: A Science of Embodied Mathematics.” Journal for Research in Mathematics Education 33.1 (2002): 59–63. JSTOR. Web.
6 Feb. 2008.
Prinz, Jesse J. Furnishing the Mind: Concepts and Their Perceptual Basis. Cambridge: MIT
Press, 2002. Print.
Quine, W.V.O. “A Postscript on Metaphor.” On Metaphor. Ed. Sheldon Sacks. Chicago: U
Chicago P, 1979. 159–60. Print.
Rakova, Marina. “The philosophy of embodied realism: A high price to pay?” Cognitive
Linguistics 13.3 (2002): 215–44. Gale Cengage. Web. 7 Nov. 2011.
Rapp, Christof. “Arisotle’s Rhetoric.” Stanford Encyclopedia of Philosophy. 2 May 2002. Web.
17 Feb. 2009.
Reddy, Michael J. “The Conduit Metaphor - A Case of Frame Conflict in Our Language about
Language.” Metaphor and Thought. Ed. Andrew Ortony. 1st ed. Cambridge: Cambridge
UP, 1979. 284–310. Print.
Resnik, Michael D. Frege and the Philosophy of Mathematics. Ithaca: Cornell UP, 1980. Print.
Richards, I.A. The Philosophy of Rhetoric. Oxford: Oxford UP, 1936. ‘‘Lecture V: Metaphor”
reprinted in Johnson, Philosophical Perspectives on Metaphor, 48–62. Print.
215
Ricoeur, Paul. The Rule of Metaphor. Trans. Robert Czerny. Toronto: U of Toronto P, 1975.
Print.
Rotman, Brian. “Gesture in the Head: Mathematics and Mobility.” Mathematics and Narrative conference, Mykonos, 2005. Thales + Friends. Web. 19 Aug. 2009.
–––. Mathematics as Sign: Writing, Imagining, Counting. Stanford: Stanford UP, 2000.
Print.
Russell, Bertrand. The Philosophy of Logical Atomism. London: Routledge, 2010. Print.
Sawyer, W.W. Prelude to Mathematics. Harmondsworth, Middlesex: Penguin, 1955. Print.
Schiralli, Martin and Nathalie Sinclair. “A Constructive Response to ‘Where Mathematics
Comes From’.” Educational Studies in Mathematics 52.1 (2003): 79–91. JSTOR. Web. 25
Feb. 2007.
Scholtz, Oliver. “‘What Metaphors Mean’ and how Metaphors Refer.” Reflecting Davidson.
Ed. Ralf Stoecker. New York: de Gruyter, 1993. 161–71. Print.
Searle, John. “A Taxonomy of Illocutionary Actions.” Expression and Meaning. Cambridge:
Cambridge UP, 1979. 1–29. Print.
–––. “Indirect Speech Acts.” Expression and Meaning. Cambridge: Cambridge UP, 1979.
30–57. Print.
–––. “Literal Meaning.” Expression and Meaning. Cambridge: Cambridge UP, 1979. 117–36.
Print.
–––. “Metaphor.” Expression and Meaning. Cambridge: Cambridge UP, 1979. 76–116. Print.
–––. Speech Acts. Cambridge: Cambridge UP, 1969. Print.
–––. “The Logical Status of Fictional Discourse.” Expression and Meaning. Cambridge: Cambridge UP, 1979. 58–75. Print.
Shakespeare, William. Romeo and Juliet. Shakespeare Online. Web. 24 Jul. 2013.
Shapiro, Stewart, ed. The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford:
Oxford UP, 2005. Print.
Shelley, Percy Bysshe. “A Defence of Poetry.” Peacock’s Four Ages of Poetry, Shelley’s Defence of Poetry, Browning’s Essay on Shelley. Ed. H.F.B Brett-Smith. Oxford: Blackwell,
1929. 23–59. Print.
“simile, n.” The Oxford English Dictionary. 2nd ed. 1989. OED Online. Oxford UP. Web. 18
Feb. 2009.
Skorupski, John. “Later Empiricism and Logical Positivism.” The Oxford Handbook of Philosophy of Mathematics and Logic. Ed. Stewart Shapiro. New York: Oxford UP, 2005.
Print.
Smith, David L. Freud’s Philosophy of the Unconscious. Dordrecht: Kluwer, 1999. Print.
Smith, Robin. “Aristotle’s Logic.” Stanford Encyclopedia of Philosophy. 23 Mar. 2011. Web.
28 Jan. 2014.
Sutton, David. Platonic and Archimedean Solids. New York: Walker and Company, 2002.
Print.
“transfer, v.” OED Online. Dec. 2011. Oxford UP. Web. 19 Feb. 2012.
Triebel, Hans. Fourier Analysis and Function Spaces. Leipzig: Teubner, 1977. Print.
Tsohatzidis, Savas L. “Speaker meaning, sentence meaning, and metaphor.” Foundations of
Speech Act Theory: Philosophical and linguistic perspectives. Ed. Savas L. Tsohatzidis.
London: Routledge, 1994. Print.
216
Turing, Alan M. “On Computable Numbers, with an Application to the Entscheidungsproblem.” Proceedings of the London Mathematical Society. Series 2. 42 (1937): 230–65. Oxford
Journals Online. Web. 13 Jun. 2012.
Vico, Giambattista. The New Science of Giambattista Vico. Trans. Thomas Bergin and Max
Fisch. New York: Doubleday Anchor, 1961. Print.
Walton, Kendall. “Metaphor and Prop Oriented Make-Believe.” European Journal of Philosophy 1.1 (1993): 39–56. Print.
–––. Mimesis as Make-Believe. Cambridge: Harvard UP, 1990. Print.
Wigner, Eugene. “The unreasonable effectiveness of mathematics in the natural sciences.”
Communications on Pure and Applied Mathematics 13.1 (1960): 1–14. Print.
Wittgenstein, Ludwig. “A Lecture on Ethics.” The Philosophical Review 74.1 (1965): 3–12.
JSTOR. Web. 2 Mar. 2009.
–––. Philosophical Investigations. Trans. G.E.M. Anscombe. 3rd ed. Oxford: Basil Blackwell,
1972. Print.
–––. Remarks on the Foundations of Mathematics. Ed. G.H. von Wright, R. Rhees, and
G.E.M. Anscombe. Trans. G.E.M. Anscombe. Cambridge: MIT Press, 1956. Print.
Yablo, Stephen. “Abstract Objects: A Case Study.” Things. New York: Oxford UP, 2010.
200–220. Print.
–––. “Does Ontology Rest on a Mistake?” Things. New York: Oxford UP, 2010. 117–44.
Print.
–––. “Go Figure: A Path through Fictionalism.” Things. New York: Oxford UP, 2010. 177–
99. Print.
–––. “Must Existence-Questions Have Answers?” Things. New York: Oxford UP, 2010. 296–
314. Print.
–––. “The Myth of the Seven.” Things. New York: Oxford UP, 2010. 221–45. Print.
–––. “Why I am Not a Nominalist.” Yablo’s personal website. 2003. Web. 9 Apr. 2013.
Zwicky, Jan. “Mathematical Analogy and Metaphorical Insight.” The Mathematical Intelligencer 28.2 (2006): 4–9. Print.
217